Thermodynamic aspects of first-order phase transformations with hysteresis in magnetic materials

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 316 (2007) 262–268 www.elsevier.com/locate/jmmm

Thermodynamic aspects of first-order phase transformations with hysteresis in magnetic materials Vittorio Bassoa,, Carlo P. Sassoa, Martino LoBueb a

Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy SATIE UMR CNRS 8029, IFR d’Alembert, ENS de Cachan, 61 Avenue du Pre´sident Wilson, 94235 Cachan, France

b

Available online 24 March 2007

Abstract In this paper we study first-order phase transformations between two stable phases with different crystal structures and magnetic order. The out-of-equilibrium thermodynamics of the hysteretic phase transformation is formulated for a collection of bistable objects by considering the two stable states as two generic phases, say 0 and 1. In the phase coexistence region we compute the magnetization, the entropy and the entropy production. Results are obtained by deriving the model parameters from experimental data on Gd-Si-Ge and giving prediction on temperature driven hysteresis, and on the heat flux with the thermal bath. r 2007 Elsevier B.V. All rights reserved. PACS: 05.70.Ce; 75.30.Sg; 75.60.Ej Keywords: First-order phase transition; Hysteresis; GdSiGe

1. Introduction A renewed interest in the thermodynamic aspects of the systems with hysteresis has been generated by the development of materials with enhanced magneto-caloric effect displaying a magneto-structural phase transformation. With the term magneto-caloric effect we refer generically to the property of a magnetic material to change its temperature (when thermally isolated) or exchange entropy with the surroundings (when in contact with an isothermal bath) when its magnetic state is changed by means of an external magnetic field. In ferromagnetic materials the change in the magnetic state has an intrinsic out-ofequilibrium character and from the thermodynamics viewpoint the presence of hysteresis calls then for a proper employ of the concepts of irreversible thermodynamics. Two facts have to be noticed in order to characterize the non-equilibrium properties of hysteretic systems. (i) Systems with hysteresis are characterized by the presence of a multiplicity of metastable energy minima in the free energy. Corresponding author. Tel.: +39 11 3919 842; fax: +39 11 3919 834.

E-mail address: [email protected] (V. Basso). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.03.177

These configurations are long-living out-of-equilibrium states. (ii) Thermodynamic irreversibility results from the fact that under an external action the system state traverses a sequence of out-of-equilibrium states. This gives rise to the spontaneous generation of entropy or entropy production. These two points do not allow to employ the classical nonequilibrium thermodynamics based on the local equilibrium hypothesis. For this theory, equilibrium is assumed to hold for small volumes only and differences from point to point give rise to fluxes of the extensive variables. This is not the case of hysteretic systems where the system can be considered homogeneous and still not at the equilibrium state. One of the most fertile fields, for the thermodynamics of systems with hysteresis, has been that of martensitic–austenitic phase transformations [1] in which it is often considered the local bistability hypothesis. This hypothesis, already present in the works of Ne´el on magnetic hysteresis in the 1940s [2], consists in subdividing the system for small elements in such a way that each single element has just two configurations in its free energy [1,3]. The system is then considered as it was a mere collection of independent bistable objects. This idea has been used in the last few

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years to study hysteresis and relaxation effects [4]. Even if proofs of the validity of such an operation do not exist, its value is sustained a posteriori by the high number of phenomena that this assumption is able to describe or to predict in quite simple ways [5]. The merit of this approach is that it is possible to describe the internal state of the system by a simple representation of the state of bistable contributions. In the rate independent case this state is determined by the time history of the controlled variable (i.e. the magnetic field H). In this paper we apply this idea to the description of a phase transformation with hysteresis. As a consequence of structural disorder the transformation is nonhomogeneous in the material and proceeds as a sequence of out-ofequilibrium states characterized by hysteresis [1]. We consider a phase transformation in the solid state between two phases, say 0 and 1, corresponding to well defined crystal structures and magnetic properties. To each phase it is associated an equilibrium Gibbs free energy, say g0 and g1 , respectively. In order to give a macroscopic description of this out-ofequilibrium phenomena we imagine to take the system in its phase-coexistence state and to make a partition into a certain number of elementary subsystems. The size of the subdivision should be such that subsystems are small enough portions of the system that are characterized by the property to be entirely either in phase 0 or phase 1 (local bistability) and large enough to still allow a thermodynamically large number of internal degree of freedoms. In this way, the role of disorder in the transformations is completely described by taking into account the statistical fluctuations of the properties from one subsystem to the other. To develop the approach we assume that the bistable objects are independent and we formulate their thermodynamics. This is developed by the following two steps.  First we suppose that intensive variables (the magnetic field H as well as the pressure p and the temperature T) are the same for all the bistable subsystems and that the Gibbs free energies of the two pure phases g0 and g1 are known functions of the extensive variables H, p and T.  Second we consider that, as a consequence of the structural disorder that makes the transformation nonhomogeneous in the material, each bistable will have small differences in the energy levels at which the transition occurs. We introduce two parameters (gu and gc ) to describe the bistable character of the unit. The transition occurs from 0 to 1 if ðg0  g1 Þ=2 ¼ gu þ gc and from 1 to 0 if ðg0  g1 Þ=2 ¼ gu  gc . The two parameters are properties of the individual subsystem and, as such, they will vary at random from subsystem to subsystem. In this way local interactions with the disorder are completely summarized in the two values gc and gu and the ensemble can be described in simple mathematical terms. The global state of the material under phase transforma-

263

tion is described by the knowledge of the state (phase 0 or phase 1) of each subsystem characterized by the couple ðgc ; gu Þ. The total Gibbs free energy per unit mass of the ensemble is the weighted sum of the energy of each unit. The result is that the units change their state because g0 and g1 are functions of the intensive variables: the temperature T, the magnetic field H, the pressure p, and so on. The history dependence of the state of the system is then generated by the history dependence of the difference g0  g1 as a function of time. As an application of the model we consider the class of materials with the so-called giant magnetocaloric effect (i.e. including the latent heat of a first-order phase transformation) [6]. For the alloy Gd-Si-Ge [7] we derived the model parameters from the experimental magnetic constitutive relation in Ref. [8]. The thermal quantities (i.e. the specific entropy exchanged De s with the thermal reservoir) can be predicted along hysteresis branch generated by changing the magnetic field H at constant temperature or by changing the temperature T at constant magnetic field. The entropy exchanged is measurable by isothermal calorimetry because TDe s is the heat flux across the surface of the sample. Then thermal experiments on magnetic materials can be compared with the model predictions presented here. This verification would permit to justify the local bistability hypothesis assumption made in the development of the approach.

2. Phase transformation with hysteresis Let us consider two phases 0 and 1 with Gibbs free energies of the pure phases g0 and g1 , respectively. We consider continuous transformations and we use the letter X for the phase fraction of phase 1. The local bistability hypothesis consists in considering the first-order phase transformation as an ensemble of several independent simple transitions. Each subsystems ðiÞ experience a transformation which is influenced by the disorder and the local order parameter is xðiÞ , the local phase fraction which is limited to assume value 0 or 1. In the following we will define the energy levels of the individual subsystem.

2.1. Individual unit Let us consider the specific Gibbs free energy gðiÞ of a given individual subsystem ðiÞ of mass mðiÞ . The two phases correspond to an energy profile gðiÞ as a function of the local order parameter xðiÞ , with two minima at xðiÞ ¼ 0 and xðiÞ ¼ 1. To simplify the notation we drop the index ðiÞ and use x to refer to the generic unit. Let us introduce the letter Z for the half difference of Gibbs free energies of the two pure phases g0 and g1 Z¼

g0  g1 2

(1)

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264

g(i)

(B)

g(i)

x

g(0) 1 gc

A

gu-Z

(1)

g(1)

A -gu+Z

(0)

0 1x

0

Z=0

Z = gu

g(i)

gu -gu

1x

0

1x

0

1x

Fig. 1. Top: profiles of the Gibbs free energy of individual unit gðiÞ as a function of its local order parameter x. Bottom: metastability of the Gibbs free energy profile gðiÞ at different values of the difference between the Gibbs potentials of the pure phases Z ¼ ðg0  g1 Þ=2. g0 and g1 , are known functions of the intensive variables: the temperature T, the magnetic field H, the pressure p, and so on.

and the letter A for the half sum: g þ g1 . (2) A¼ 0 2 The relevant energies of the profile of the unit are defined as gð0Þ ¼ A þ ðZ  gu Þ,

ð3Þ

gð1Þ ¼ A  ðZ  gu Þ,

ð4Þ

g

ðBÞ

¼ A þ gc , ð0Þ

ð5Þ ð1Þ

where g and g are the energy levels of the state 0 and 1 of the unit, and gðBÞ is the energy of an intermediate unstable state. The profile is illustrated in Fig. 1 top. In Fig. 1 bottom, the picture at Z ¼ 0 shows that the role of gu is to introduce a shift in the energy of the stable states as a consequence of the disorder. The condition for the transformation is that it occurs by the passage through an intermediate unstable state of energy gðBÞ ¼ A þ gc with gc 40. Then the transformation occurs when gð0Þ ¼ gðBÞ g

ð1Þ

¼g

ðBÞ

for 0 ! 1,

ð6Þ

for 1 ! 0.

ð7Þ

This condition leads to switching condition for Z: Z ¼ gu  gc ,

gu

Z

gu

Z

Fig. 2. Left: order parameter x of the individual unit switching from 0 to 1 and from 1 to 0 at threshold values of the variable Z ¼ gu  gc . Right: hysteretic behavior of the energy assumed by the individual unit. The open symbols corresponds to the three conditions Z ¼ 0, Z ¼ gu and Z ¼ gu þ gc shown at the bottom of Fig. 1.

Z = gu + gc

g(i)

g(i)

0

gc

gc

(8)

where the þ sign refer to the 0 ! 1 and  to the 1 ! 0 transitions (see Fig. 2). In Eq. (8) the difference Z ¼ ðg0  g1 Þ=2 refers to a perfectly homogeneous medium and is the same for all

bistables while gc and gu are a consequence of the local structural disorder and refer to a specific unit. Then a unit changes its state because g0 and g1 , are functions of the intensive variables: the temperature T, the magnetic field H, the pressure p, and so on. The history dependence of the state (0 or 1) of the subsystem is then generated by the history of the difference Z ¼ ðg0  g1 Þ=2 as a function of time that takes the role of an external driving force. On the right-hand side of Eq. (8) the term gu  gc describes the effect of structural disorder in the given portion. The values of gu and gc are properties of the individual subsystem and, as such, they will vary at random from subsystem to subsystem according to some statistical distribution characterizing the disorder in the material. We suppose here that gu and gc are independent of the intensive variables. 2.2. Dynamics of the unit The assumption made for the energy levels of the unit permits to discuss the time scales over which the transition takes place. Two relevant time scales exists. Switching dynamics: The switching from 0 to 1 and 1 to 0 is connected to an intrinsic damping. As in Ref. [1] we can write that locally the switching rate is proportional to a force given by the gradient of the free energy with respect to the independent variable x: dx qgðiÞ ¼ , (9) dt qx where g is the damping coefficient. The kinetic depends therefore on the specific shape of qgðiÞ =qx. Since we have not assumed any specific shape, we may estimate the typical time for the transition to occur. At Z ¼ gu þ gc the switching from 0 to 1 occurs in the time Dt. We can approximate Eq. (9) as g

Dx DgðiÞ ¼ (10) Dt Dx obtaining then Dt ¼ g=ð2gc Þ proportional to inverse of the energy barrier gc . g

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Relaxation dynamics: Thermal relaxation effects are connected to the probability that the transitions occur spontaneously by thermal activation over the barrier. The master equation for the transition is dP1 ¼ w01 P0  w10 P1 , dt

(11)

where P0 and P1 are the probability to find the system state in 0 and 1, respectively. The transition rates w01 and w10 are given by exponential of the energy barrier to overcome (gðBÞ  gð0Þ from 0 to 1 and gðBÞ  gð1Þ from 1 to 0) over the thermal energy kB T. We obtain the rates as " # 1 gc  Z þ gu , ð12Þ w01 ¼ exp  t0 gf " # 1 gc þ Z  gu w10 ¼ exp  , ð13Þ t0 gf where gf ¼ kB T=m gives the intensity of the thermal fluctuations. m is the mass of the units and is taken as a constant for all units. t0 is a time constant connected to intrinsic fluctuation times. The situation is similar to that of thermodynamics of magnetic bistables of Ref. [5]. As for that case, instead of observing the probability of each bistable ðgc ; gu Þ, it is possible, if kB T is not too large, to observe the relaxation effects around a time dependent line in the plane ðgc ; gu Þ. The derivation is given in Ref. [5]. Rate independent limit: In the following we consider, without losing generality, that the switching dynamics take place on a time scale of orders of magnitude shorter then the rate of change of the input variable Z and that relaxation effects takes place over a time scale which is orders of magnitudes larger then the typical rate of Z. Both dynamics can be disregarded and hysteresis effects can then be studied in the rate independent limit. 2.3. Superposition The global state of the material under phase transformation is described by the introduction of the function xðgc ; gu Þ, with the property that x ¼ 0 or 1 depending on whether the subsystem characterized by the values ðgc ; gu Þ is in phase 0 or phase 1, respectively. The total Gibbs free energy per unit mass of the material can be expressed as Z 1 Z 1 g¼ dgc ½ð1  xðgc ; gu ÞÞgð0Þ 0

1

þ xðgc ; gu Þgð1Þ  pðgc ; gu Þ dgu ,

ð14Þ

where pðgc ; gu Þ is the probability density that a given subsystem be characterized by given values of ðgc ; gu Þ. The time history of Z ¼ ðg0  g1 Þ=2 determines which elementary portions of the system and in which phase. The switching rules turn out to be identical to the rules governing the Preisach model of hysteresis [5] (see Fig. 3). All the methods developed for that model can be immediately applied to the present phase-transformation

Fig. 3. History dependence of the bðgc Þ line. Left: example of a time evolution of the difference between the Gibbs free energy ZðH; TÞ ¼ ðg0 ðH; TÞ  g1 ðH; TÞÞ=2 of the two phases due to changes in H and T. Right: corresponding regions of the plane ðgc ; gu Þ in the phase 1 or 0. The borderline bðgc Þ is determined, as in the Preisach model of hysteresis, by enforcing the inequality jbðgc Þ  ZðtÞjpgc at each instant of the time history.

model and the Gibbs free energy can be written in the form: "Z Z 1 bðgc Þ dgc ðgu  ZÞpðgc ; gu Þ dgu g¼Aþ 0

Z

1

1

#

ðgu  ZÞpðgc ; gu Þ dgu ,



ð15Þ

bðgc Þ

where the function bðgc Þ represents the boundary between the regions in the ðgc ; gu Þ plane associated with elementary portions in phase 0 or phase 1. The bðgc Þ line fully characterizes the nonequilibrium phase-coexistence state of the material. Given the time history ZðtÞ, the corresponding state line bðgc Þ is determined by enforcing the inequality jbðgc Þ  ZðtÞjpgc for all values of gc at each instant of the time history (see Ref. [5] for a detailed discussion of this evolution rule). The total mass fraction X of the material which is in phase 1 is given by Z 1 Z bðgc Þ X¼ dgc pðgc ; gu Þ dgu , (16) 0

1

where the previous integral expression on the plane ðgc ; gu Þ is analogous to that of the Preisach model of hysteresis [5]. 2.4. State variables and internal variables From the analysis of the previous section we see that the state of the system is completely described by the historydependent state line bðgc Þ. The state-line takes the role of internal thermodynamic variable not explicitly coupled to intensive variables. We then follow the approach of Ref. [5]. Let us consider as extensive variables the magnetization M and the specific entropy s (per unit mass) and the corresponding conjugated intensive variables, magnetic field H and temperature T. M and s are given by  qg  M¼  , (17) qH T;b

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266

 qg  s¼  qT H;b

(18)

and can be calculated by the knowledge of the distribution pðgc ; gu Þ. We suppose here that the distribution pðgc ; gu Þ does not depend on H and T, Eq. (15) gives M ¼ XM 1 þ ð1  X ÞM 0 ,

(19)

s ¼ Xs1 þ ð1  X Þs0 ,

(20)

where M 1 ¼ qg1 =qH, M 0 ¼ qg0 =qH, s1 ¼ qg1 =qT, s0 ¼ qg0 =qT and X is the phase fraction given by the Preisach model expression Eq. (16) with ZðH; TÞ as input. The rate of entropy production di s=dt is defined as Z 1  di s qg qb ¼ dg T (21) dt qbH;T qt c 0 giving di s T ¼2 dt

Z

1 0

low-temperature ferromagnetic GdSi-type structure (phase 1) and a high-temperature anti-ferromagnetic SmGe-type structure (phase 0) at T ’ 70 K [7]. In a previous study [8] the model that describes the magnetization curves of the material was determined. The Gibbs free energy of the two phases are approximated as 2 g0 ¼ f 0 ðTÞ  m0 ðM ð0Þ 0 H þ w0 H =2Þ,

g1 ¼ f 1 ðTÞ 

m0 M ð0Þ 1 H,

ð24Þ

where f 0 ðTÞ and f 1 ðTÞ are the field-pressure-independent parts of the free energy and M 0 ¼ M ð0Þ 0 H þ w0 ,

ð25Þ

M ð0Þ 1

ð26Þ

M1 ¼

are the magnetization curves. The constants w0 , M ð0Þ 0 and M ð0Þ are normalized per unit mass. We have then 1 ð0Þ 2 Z ¼ 12½Df ðTÞ  m0 ððM ð0Þ 0  M 1 ÞH þ w0 H =2Þ

qb ½Z  bðgc Þ pðgc ; bðgc ÞÞ dgc . qt

(22)

A material that can be described by this approach is the alloy Gd5 ðSix Ge1x Þ4 . This compound (with x ¼ 0:082) undergoes a first order phase transformation between a

M (Am2kg-1)

magnetization

T = 80 K

μ0 H = 2 T

magnetization

entropy

T = 80 K

μ0 H = 2 T

entropy

150 100 50

s - s0(Jkg-1K-1)

0 0.00 −0.10 −0.20 0.010 Δis (Jkg-1K-1)

(28)

where Df ¼ f 0  f 1 and f A ¼ f 0 þ f 1 . The hysteresis loops of Gd5 ðSix Ge1x Þ4 , with x ¼ 0:082, and their temperature dependence are obtained by specifying the values of the parameters. We take the values obtained in Ref. [8]: ð0Þ 2 1 2 1 M ð0Þ and w0 ¼ 9:42 1 ¼ 200 Am kg , M 0 ¼ 11 Am kg 1 6 3 10 m kg . In the range T ¼ 68290 K, Df ðTÞ is

250 200

(27)

and ð0Þ 2 A ¼ 12½f A ðTÞ  m0 ððM ð0Þ 0 þ M 1 ÞH þ w0 H =2Þ

3. Phase transformation driven by magnetic field or temperature

ð23Þ

entropy production

entropy production

0.005 μ0H = 2 T

T = 80 K 0.000 0

2 4 6 Magnetic field μ0H (T)

8

60

70

80

90

Temperature T (K)

Fig. 4. Application of the model to the phase transformation of GdSiGe material. The six pictures display: magnetization M, entropy change s  s0 (s0 is the entropy of the phase 0 taken as reference) and the integral of the entropy production Di s as a function of the magnetic field m0 H at constant temperature T ¼ 80 K and of the temperature T at constant magnetic field m0 H ¼ 2 T. With the parameters used (see text) the M H loop (top-left) reproduces the experimental behavior of Gd5 ðSix Ge1x Þ4 with x ¼ 0:082 of Ref. [7]. The thermal quantities (s and Di s) as a function of H and all the loops as a function of the temperature are predictions of the model without adjustable parameters.

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150

A

(29)

with sc ¼ 2:5 J kg1 , su ¼ 30 J kg1 , and gc0 ¼ 27 J kg1 . The entropy and the entropy production are given by expressions (Eqs. (20) and (22)) containing the same parameters that control the shape of magnetic hysteresis loops, hence predictions of hysteresis as a function of T and of the entropy can be made without any additional adjustable parameter. The magnetization, the entropy and the entropy production were calculated by using the same parameters as a function of the magnetic fields (m0 H is cycled between 0 and 7 T at constant temperature T ¼ 80 K) and as a function of the temperature (T is cycled between 60 and 90 K at constant magnetic field m0 H ¼ 2 T). The results are shown in Fig. 4. A measurable calorimetric quantity is the heat flux TDe s that is transferred from the thermal reservoir to the sample in order to keep temperature constant. The net entropy De s transferred between the thermal reservoir and the system is given by the integral over the transformation of the infinitesimal entropy balance equation de s ¼ ds  di s, where ds is the entropy change and di s the entropy production contribution. Therefore we define De s ¼ Ds Di s, where Ds ¼ s  s0 is given by Eq. (20) while the internal entropy production Di s is given by the integral of Eq. (22) over the transformation. Fig. 5 shows an example of the computed behavior of De s in the internal hysteresis branching. In order to obtain a larger ratio between the entropy production and the entropy change the example is computed with the same parameter given previously apart from sc ¼ 50 J kg1 and dDf =dTjT 0 ¼ 4:6 J kg1 K1 . 4. Conclusions and future extensions We have developed here a phenomenological approach to the thermodynamics of hysteresis in a first-order phase transition. The approach has the merit to give a clear definition of the internal state of the system and to give predictions of measurable quantities, the magnetization M, the entropy s and the entropy production along the irreversible transformations. The role played by the intensive variables (magnetic field H, temperature T, and so on) in driving the transformation turns out to be equivalent because they appear in the expressions of the difference of the Gibbs free energies g0  g1 of the two phases which is the input variable of the hysteresis model. The price to pay in our present development is that the thermodynamics of the pure phases has to be known in

C M (Am2kg-1)

with T 0 ¼ 80 K, Df ðT 0 Þ ¼ 640 J kg1 and 1 1 dDf =dTjT 0 ¼ 46 J kg K . In addition the distribution is " #   ðgc  gc0 Þ2 g2u pðgc ; gu Þ / exp  (30) exp  2s2c 2s2u

E 100 F D B 50 2.5

3.0

3.5 4.0 Magnetic field μ0H (T)

4.5

2.10−3 A 0 C Δes (Jkg-1K-1)

approximated by a linear function of temperature:  dDf  ðT  T 0 Þ Df ðTÞ ¼ Df ðT 0 Þ þ dT T 0

267

E

–2.10−3 F

−4.10−3

D

−6.10−3

−8.10−3

B

2.5

3.0

3.5

4.0

4.5

Magnetic field μ0H (T) Fig. 5. Example of magnetization curves and exchanged entropy in internal hysteresis branching. Top: the magnetic field is changed at constant temperature in the sequence A; . . . ; F and back to A. Bottom: the net entropy exchanged with the reservoir increases due to internal entropy production.

advance. g0 and g1 are in fact supposed to be known. Since experiments are needed for their determination, it has to be possible to perform experiments by fixing the material in one or the other phase. This fact may create some difficulties for the applications of this approach to real cases. A crucial point of this phenomenological approach is that we consider that temperature is always a controlled parameter. This may not be always the case. Let us consider for example adiabatic processes. In that case the temperature of the system is determined by the thermal capacity of the body itself. The whole approach is still valid if we consider that a real material is also composed by other degrees of freedom (lattice vibrations, electrons, and so on) that can be always considered as in equilibrium. The temperature is then determined by the equilibrium part of the material that is acting as a thermal reservoir of finite heat capacity and the hysteresis model describes only the degrees of freedom that are out-ofequilibrium. However if the thermal capacity of the

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thermal reservoir is too small, the use of an out-ofequilibrium approach like the one presented here is problematic and the whole out-of-equilibrium approach cannot be introduced because the temperature of the system cannot be defined. The only assumption made in the development of the approach is that the hysteretic process is related to the phase transformation and not with the magnetization processes inside the phases. This is the case of magnetocaloric materials involving a phase change. A number of model predictions can then be applied to this class of materials. Furthermore, the approach developed here may be easily extended to include dynamic effects in giant magnetocaloric materials [9]. The model itself may be also extended by introducing the dependence of the bistable parameters on intensive quantities giving more refined

description of the thermal quantities. This will be the direction for future investigations. References [1] [2] [3] [4] [5]

[6] [7] [8] [9]

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