Calculation model on magnetocaloric effect (MCE) and relative cooling power (RCP) in composite materials at room temperature

Journal of Magnetism and Magnetic Materials 449 (2018) 500–504

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Calculation model on magnetocaloric effect (MCE) and relative cooling power (RCP) in composite materials at room temperature Moulay Youssef El Hafidi ⇑, Abderrazzak Boubekri, Mohamed El Hafidi Laboratory of Condensed Matter Physics (LCMP), Department of Physics, Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, Cdt Driss El Harty Avenue, BP 7955, Sidi Othmane, 20645 Casablanca, Morocco

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 20 April 2017 Received in revised form 21 September 2017 Accepted 23 October 2017

In this paper, a calculation model is proposed to measure magnetocaloric effect (MCE) and relative cooling power (RCP) in composites based on magnetic materials underscoring a giant MCE at room temperature. The two composite materials targeted are Gd-Gd5Si2Ge2 and MnAs-Mn1þd As0:9 Sb0:1 due to their high magnetic entropy change DSM over 270–300 K and 280–320 K. Our selected composites could give a larger RCP value among existing magnetocaloric materials for magnetic refrigeration in the temperature range of 280–300 K which is desirable for ideal Ericssoncycle magnetic refrigeration. The excellent magnetocaloric properties of these two magnetic composites make them attractive for active magnetic refrigeration at room temperature. Ó 2017 Elsevier B.V. All rights reserved.

Keywords: Spin Hamiltonian Magnetic refrigeration Magnetocaloric effect Composite materials

Contents 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . Model . . . . . . . . . . . . . . . Thermodynamic study . . Results and discussion . . Conclusion and outlook . References . . . . . . . . . . .

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1. Introduction Magnetic refrigeration is a clean technology that uses a magnetic field to change the magnetic entropy of a material (the magnetocaloric effect, MCE), thus allowing the material to serve as a refrigerant. This technology provides a higher cooling efficiency (about 20–30%) than conventional gas compression techniques [1]. Many efforts have been spent both theoretically and experimentally in order to formulate new concepts of Active Magnetic Refrigeration (AMR) which permit magnetic refrigeration at room temperature (see as example Ref. [2]). The aim being to substitute the conventional refrigeration releasing greenhouse gases by a new ⇑ Corresponding author. E-mail address: [email protected] (M.Y. El Hafidi). https://doi.org/10.1016/j.jmmm.2017.10.083 0304-8853/Ó 2017 Elsevier B.V. All rights reserved.

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500 501 501 502 504 504

technology preserving the environment and saving the energy consumption. Producing a magnetocaloric material with a large magnetic entropy change DSM over a wide temperature range, i.e., a large refrigerant capacity, is of capital interest for magnetic refrigeration applications. For ideal Ericsson cycle based magnetic refrigeration, a magnetocaloric material should possess a constant magnetic entropy change DSM in the refrigeration temperature range [3]. Our ultimate objective is to work out a model providing magnetocaloric effect (MCE) and relative cooling power (RCP) calculations in magnetic composites materials. Indeed, several theoretical approaches and simulations have been used to predict and describe the MCE in ideal materials (see as examples Refs. [4]; [5]) while experimentally, since the discovery of giant MCE in Gd5Si2Ge2 in 1997 [6], a vast variety of materials has been synthesized

M.Y. El Hafidi et al. / Journal of Magnetism and Magnetic Materials 449 (2018) 500–504

and the MCE field surged up. This progress covered pure magnetic elements and their solid solution alloys such as 4f elements (Gd, Nd, . . .), binary and intermetallic compounds such as R-M2 materials (R = Dy, Ho, Er; M = Co, Al. . .), Gd5(Si1-xGex)4 and related materials that exhibit giant MCE, families of Mn-based compounds, La (Fe13xMx)-based compounds and manganites (La1-xMx) MnO3 where M = Na, Ag with quite important MCE (for a review see Ref. [7]). For illustration, we choose the two composites Gd-Gd5Si2Ge2 and MnAs- Mn1þd As0:9 Sb0:1 because of their high DSM over 270– 300 K and 280–320 K. The paper is organized as follows: in Section 2, we introduce the model by giving the spin Hamiltonian and partition function. Then in Section 3, we calculate relevant thermodynamic parameters influencing the MCE. Section 4 is devoted to numerical results and discussions. Finally, we give conclusion and outlook. 2. Model We first consider the Heisenberg Hamiltonian of a ferromagnetic asingle-system given by

X z X Ha ¼ J a ~ Sj;a  g lB l0 H0  Si;a Si;a :~ hi;ji

ð1Þ

ing on the site i (respectively j) and H0 is the applied magnetic field parallel to the z-axis of spin. Here, g is the Landé factor, lB is the Bohr magneton and l0 is magnetic permeability. In order to solve Eq. (1), we make an approximation by defining an effective molecular field at the th

i

site : BW ¼

1 X hSj iJ ij

g lB j2nn

ð2Þ

Therefore, the exchange interaction is replaced by the effective molecular field BW produced by the neighboring spins. We are now able to treat this problem as if the system were a simple paramagnet placed in a magnetic field B + BW. The effective Hamiltonian can now be written as:

Ha ¼ g lB

X Si;a ðB0 þ BW Þ

ð3Þ

i

The assumption supporting this approach is that all magnetic ions experience the same molecular field. However, this may be rather questionable, particularly at temperatures close to a magnetic phase transition. For a ferromagnet, the molecular field will act so to align neighboring magnetic moments. This is because the dominant exchange interactions are positive. Since the molecular field measures the effect of the system ordering, we can assume that BW = k M where:

k¼

zJ ðg lB Þ2 n

ð4Þ

Za ¼

Y

ð6Þ

Z i;a

i

where

Z i;a ¼

mX i ¼þS X ebglB mi ðB0 þBW Þ ¼ eymi

ð7Þ

mi ¼S

fmi g

y ¼ bg lB ðB0 þ BW Þ ¼ bg lB ðB0 þ kMÞ; b = 1/kBT, kB being the Boltzmann constant. By performing the Za calculation, we deduce the Helmholtz free energy for n spins per unit volume using the expression Fa = n kB ln(Za). Thus, the magnetization, the magnetic entropy per unit volume and the magnetic specific heat are expressed respectively by:

MðB; TÞ ¼ M S BS ðyÞ

ð8Þ

    1 sh 1 þ 2S y  yBS ðyÞ SM ¼ nkB ln y sh 2S

ð9Þ

" C M ¼ nkB y

2

 1 2 # 1 2 ð1 þ 2S Þ 2S   2   1  2 sh 2Sþ1 sh 2S y y 2S

ð10Þ

where Bs(y) is the Brillouin function and Ms < =ce : inf >¼ ng lB S is the saturation magnetization. Note that in absence of both exchange (paramagnetic system) and external magnetic field, one finds that the entropy of the assembly of magnetic moments has its maximum at M = 0, and its value is SM (0) = nkB ln (2S + 1), corresponding to the upper limit for the entropy associated with the atomic magnetic moments with 2S + 1 discrete levels. However, in our system where exchange is taken into account, the Eqs. (8)–(10) contain implicitly the magnetization M making hard the analytical resolution of the problem. Thus, in order to overcome this difficulty and perform a numerical resolution, we have developed a computational program simulating the evolution of these variables with both magnetic field and temperature for a given set of microscopic parameters such as exchange interaction or spin size. 3. Thermodynamic study For the characterization of the magnetocaloric response of a material, three main parameters can be studied: the isothermal magnetic entropy change, DSM; the adiabatic temperature change, DTad; and the relative cooling power, RCP [8]. In order to calculate these parameters, we start form the Gibbs free energy G which is expressed in terms of internal energy U, extensive variables (entropy S and magnetization M) and intensive variables (temperature T and magnetic induction B0) as follows:

~ ~ GðT; P; BÞ ¼ U þ PV  TS  M B

ð11Þ

Thus:

Thus, k characterizes the strength of the molecular field as a function of the magnetization (for a ferromagnet, k > 0). Here z and n denote respectively the coordination number and the volume spin concentration. Both are related to the crystallographic structure of the material. In this context, our Hamiltonian may be equally rewritten as:

X Ha ¼ Ha;i

Since all individual Hamiltonians Hi, a commute between them, the partition function becomes:

i

where Ja is the exchange interaction constant between nearestneighbors (Ja > 0), ~ Si;a (respectively ~ Sj;a ) are the spin operators act-

501

ð5Þ

i

where Hai ¼ g lB Sia ðB0 þ BW Þ whose the eigenvalues are given by Ei ¼ g lB mi ðB0 þ BW Þ where S  mi  S.



dG ¼

   @G @G dT þ dB ¼ MdB  SM dT @T @B

ð12Þ

Where:

    @ @G @ @G ¼ @B @T @T @B

ð13Þ

By using Maxwell-Weiss relation [3], we get:

    @M @SM ¼ @T @B

ð14Þ

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M.Y. El Hafidi et al. / Journal of Magnetism and Magnetic Materials 449 (2018) 500–504

In the case of a reversible transformation, the differential entropy SM can be written as follows:

dSM ¼

  CM @M dT þ dB T @T

ð15Þ

where CM is the heat capacity at constant induction B. In the case of an adiabatic transformation, the magnetocaloric effect (MCE) is expressed by:

Z MCE ¼ DT ¼ 

  T @M dB C M @T

ð16Þ

Since for an isothermal process, the temperature does not change during the magnetization, we can write:



dSM ðB; TÞ ¼

@SM @B



dB

ð17Þ

T

By using Eq. (14), magnetic entropy variation can then be established for an isothermal demagnetization process, when the magnetic field changes from B0 to B1 at a given temperature as:

Z

DSM ¼

B1

B0

  @M dB @T B

ð18Þ

  where @M is thermal magnetization for a fixed magnetic field. @T B The relative cooling power (RCP) has been used most of the time as criteria for magnetic refrigeration materials [9]. The RCP is defined as:

RCPðB0 ! B1 Þ ¼ DSM max ðB0 ! B1 Þ  DT1=2 ðB0 ! B1 Þ

ð19Þ

where DSM max(B0 ? B1) and DT1/2(B0 ? B1) are the maximum value and the full width at half maximum of DSM (B0 ? B1) at given B0 and B1, respectively. Therefore, RCP is a practical way to measure the amount of heat transferred between the hot and cold reservoirs in an ideal refrigeration cycle. Thus, magnetic materials with a large RCP exhibit a large isothermal magnetic entropy change over a wide temperature range. The obtained results are confirmed with some known experimental results as for Gadolinium where Tc = 293.4 K, g = 2, S = 7/2 and which crystallizes in hexagonal structure [10].

for T < Tc, the magnetization values are quite stable at low temperatures and at T = Tc the magnetization exhibits a continuous behavior but it is not continuously differentiable. Thus, the phase transition is of second order. Temperature dependence of the relative magnetization M/MS for a given spin system under different reduced magnetic fields B/B0 is displayed in Fig. 2. Since the considered system is ferromagnetic, a non-vanishing magnetization persists even above the critical temperature when an external magnetic field is applied which tells that spins in different positions maintain their correlation and the transition is pushed back to high temperatures. The magnetic entropy curves versus temperature under different external magnetic fields for a given spin system is showed in Fig. 3. The entropy changes as function of temperature, derived from magnetic isotherms through the Maxwell relations (18) are displayed in Fig. 4 for S = 1/2. The absolute value |DSM| reaches a maximum around Tc, under various magnetic fields. This maximum increases strongly with the applied magnetic field. |DSM| is also displayed for different values of the system spin. As shown in Fig. 5, |DSM| grows significantly with the spin size. Actually, DSM is negative in the entire temperature range and is extended over a wide range of temperature around the Curie temperature, which is useful for a room temperature magnetic refrigeration process. This result is consistent with those obtained experimentally in Ref. [11]. Producing a magnetocaloric material that possesses a large magnetic entropy change |DSM| over a wide temperature range (DT), i.e., a large relative cooling power (RCP), is highly required for magnetic refrigeration applications. Therefore, both large

4. Results and discussion Temperature dependence of the spontaneous magnetization for different values of spin S is shown in Fig. 1. For simplicity, we have plotted the relative magnetization M/MS versus the reduced temperature T/Tc. Despite the slightly different shape of the curves, a general trend is present: for T > Tc, the magnetization vanishes (M = 0);

Fig. 2. Relative magnetization as a function of T/Tc for different values of reduced magnetic field.

Fig. 1. Relative magnetization as a function of reduced temp. T/Tc for different values of spin S.

Fig. 3. Magnetic entropy curves versus temperature under different external magnetic fields.

M.Y. El Hafidi et al. / Journal of Magnetism and Magnetic Materials 449 (2018) 500–504

503

Fig. 4. Absolute value of |DSM| as a function of temperature for different values of magnetic field.

Fig. 7. Magnetic entropy change of MnAs and Mn1þd As0:9 Sb0:1 [12] in comparison with Gd [13] and Gd5Si2Ge2 [14].

Fig. 5. Temperature dependence of magnetic entropy change for different values of spin.

Fig. 8. Magnetic entropy change of composite material for an applied magnetic field of 1.5T.

Fig. 6. Field dependence of RCP of a spin-1 compound.

isothermal entropy and large adiabatic temperature changes are required for good magnetic refrigeration materials> (see Fig. 6). Finally, we consider a multi-phase material with two kinds of spin in order to enhance the RCP, i.e. enhancing |DSM| and DT/1/2. This protocol would be able to describe the magnetic refrigeration efficiency of the two kinds of ferromagnets as in composite materials shown in Fig. 7. P The corresponding Hamiltonian rewrites as H ¼ a Ha where 0 Ha ] = 0, the partition function becomes then [Ha, Q ZðT; H0 Þ ¼ a Za ðT; H0 Þ. Consequently, one can add the extensive variables as DSMa. A simple composite material that we have considered is viewed as

Fig. 9. Magnetic entropy change of composite material for an applied magnetic field of 5T.

two superposed mediums with spins Sa and Sb and respective concentrations x = 0.6 and 1  x = 0.4. Figs. 8 and 9 illustrate the temperature dependence of the magnetic entropy change for the formed composite under two applied fields. Obtained results confirm that interactions between multiphase magnetic materials enhance the MCE of the formed composite and shift the peak temperatures of preliminary phases to higher temperatures [15,16]. The large temperature span and RCP values

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previewed with mixed spin systems in this work make them suitable for potential use in magnetic refrigeration and motivates the experimental investigations in this way. Our main results fit closely with conclusions proposed by Taguchi & al. in Ref. [17]. Indeed, they stipulate that both the competition and cooperation between different magnetic interactions promote the first-order nature of the magnetic transition, leading to giant magnetocaloric effects. Therefore, taking advantage of diverse instabilities is a promising strategy for exploring new magnetocaloric materials. 5. Conclusion and outlook In this article, an attempt to enhance the magnetocaloric effect and refrigerant cooling power using composite materials is viewed from the perspective of multiple spin interactions. Starting from the Heisenberg Hamiltonian of a ferromagnetic asingle-system, a calculation model is performed using the Mean Field approximation and key thermodynamic parameters are studied and simulated numerically using a Fortran program. We were exploring new composite materials that exhibit giant magnetocaloric effects to achieve sufficient cooling efficiency. Combining different kinds of spins in composite materials is very promising to obtain a large magnetocaloric effect accompanied by a strong refrigerant cooling power.

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