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Chelyabinsk Physical and Mathematical Journal. 2023. Vol. 8, iss. 2. P. 280-291.
DOI: 10.47475/2500-0101-2023-18211
GENERALIZED MODEL OF THE MAGNETOSTRUCTURAL PHASE TRANSITION IN La(Fe,Si)i3 COMPOUNDS UNDER THE SIMULTANEOUS ACTION OF TEMPERATURE, MAGNETIC FIELD AND PRESSURE
R^. Makarin1'", М.V. Zheleznyi23b, D.Yu. Karpenkov12c
1 Lomonosov Moscow State University, Moscow, Russia 2National University of Science and Technology "MISIS", Moscow, Russia 3Baikov Institute of Metallurgy and Materials Science, Russian Academy of Sciences, Moscow, Russia
"makarin.ra16@physics.msu.ru, bmarkiron@mail.ru, ckarpenkov.dy@misis.ru
The paper presents a phenomenological generalized model of a first-order metamagnetic phase transition for the La(Fe,Si)i3 compounds in the approximation of localized moments under the simultaneous action of temperature, field and pressure. To achieve the maximum cooling power of magnetic solid-state cooling devices, the Curie temperature of the working bodies should be fine-tuned by two external generalized forces: a magnetic field and pressure. The thermodynamic phenomenological models presented in the literature are mostly focused on the description of the behavior of magnetocaloric materials in the vicinity of phase transition in the absence of external pressure. In turn, the latter provide a significant reduction in the field hysteresis effect by shifting the Curie temperature and expand the working temperature range of the refrigerant. To estimate the required pressure value, a new generalized model was developed that excludes the linear dependence of the phase transition temperature on the volume change and modernizes the form of the magnetic and phonon entropy, taking into account anharmonism. In addition, the equations of state describing the behavior of working bodies underwent a multistimuli cooling cycle were obtained. The model allows estimating the upper limit of the temperature and field hysteresis and predicting the required external pressure to reduce the field hysteresis.
Keywords: magnetocaloric effect, Brillouin function, phonon entropy, phenomenological model, itinerant metamagnetic transition.
Introduction
The current century is the century of green, energy-saving technologies. In developed countries, almost a third of all generated energy is utilized on various cooling systems [1], most of which operate on the basis of vapor compression systems. Among the wide variety of alternative cooling methods [2], solid-state cooling based on the multicaloric effect is the most promising.
Over the past twenty years, the magnetocaloric effect [3; 4] has been most fully studied experimentally and theoretically; dozens of prototypes [5] and hundreds of models [6] of magnetic solid-state refrigerators have been created. The use of multistimuli cooling cycles is a new approach that requires the use of external pressure. The latter
The work was supported by the Russian Science Foundation grant No. 21-72-10147 and Russian Ministry of Science and Education grant No. 075-15-2021-1353.
makes it possible to reduce the negative effect of thermal and field hysteresis [7-10] on the efficiency of magnetic heat pumps. The decrease in hysteresis losses is ensured by the shifting of the Curie temperature by the external pressure, which can also lead to an increase in the sharpness of the phase transition [11].
The maximum magnitude of the magnetocaloric effect (giant magnetocaloric effect) is achieved in materials with a first-order magnetic phase transition. The developed phenomenological models of a first-order metamagnetic transition use both the approximation of localized moments [12-15] and the approximation of itinerant electrons [16; 17]. In particular, the models describing the behavior of the La(Fe,Si)i3 compounds are built in the framework of localized moments approximation.
Moreover, the currently existing phenomenological theories use a linear dependence of the Curie temperature on pressure, which significantly narrows the working pressure range. The shortcomings of such models include the linear dependence of the exchange energy on volume change (w) introduced by Bean and Rodbell [18]. Some improvements were made by Valiev [19] by introducing a quadratic dependence. However, the coefficients in the exchange integral vs. w equation were found from the fitting of the experiment [19].
In this article, a new generalized model was developed that excludes the linear dependence of the phase transition temperature on the volume change, and modernizes the form of the magnetic and phonon entropy, taking into account anharmonism. In addition, the equations of state describing the behavior of working bodies underwent a multistimuli cooling cycle were obtained. The model allows estimating the upper limit of the temperature and field hysteresis and predicting the required external pressure to reduce the field hysteresis.
1. Phenomenological model of the first order phase transition in La(Fe,Si)i3 compounds
The developed phenomenological model of the first-order metamagnetic phase transition for the La(Fe,Si)13 alloy uses a thermodynamic approach within the approximation of localized magnetic moments. Despite the fact that the magnetic properties of the La(Fe,Si)13 compounds are governed by itinerant 3d electrons, if we assume that the electron has only a spin moment S, then the total quantum mechanical moment on the Fe atom becomes J = 1/2 (as in the model of Bean and Rodbell [18]) and the spontaneous magnetization will be proportional to the difference in the level populations [20].
The state of such a system is described by the volume-normalized Helmholtz free energy [F] = [m?]: F = Fmag + Fph + Fel. In the equation for the thermodynamic potential, we make the substitution MM0 = mM0, where M0 = n0^BgJ saturation magnetization at zero temperature. In this case, the Helmholtz free energy can be written as follows:
F(T,p, H,w,m) = HM0m
no kB Tc (p,w)m
2
1.1. Magnetic entropy
In this chapter, we elaborate on the expression for the magnetic entropy. The first step is to write down the entropy of an ideal paramagnetic gas made up of identical non-interacting particles. We get the following expression:
Sj(T, H) = nkJ ln (Zj(T, H}) - m(H, T)B-1 (m(T, H))
where ZJ (T,H)
sinh( J xj (T,H)) .
is a partition function, xJ(T,H)
JjH, and
kB T '
B-1
sinh( j xj (T,H))
j (m(T,H)) is the inverse Brillouin function.
To take into account the exchange interaction of localized magnetic moments, we implemented the "effective" magnetic field Heff. We can do this in several ways: 1) the Weiss molecular field; 2) the Oguchi atom pair method; 3) a pair of atoms in the constant bond approximation; 4) the Rushbrook — Wood method, etc. We choose the first method in order to remain within the formalism of the Bean — Rodbell model. Then we postulate that Heff = H + AMs, where A is the Weiss molecular field constant, Ms is the spontaneous magnetization. After substituting into (2) the effective Weiss field, we obtain that
sh( ^ B-1(m))
SJ = nkB < ln
sh (JB-V))
— mB-1 (m)
3)
1.2. Lattice entropy
To obtain a modified expression for the lattice entropy Sph, it is necessary to take into account the harmonic and anharmonic contributions.
We begin by considering phonons in the harmonic approximation. There are N phonons, each of which has 3 degrees of freedom; then, in the harmonic approximation, in the case of small amplitude of vibrations, they can be viewed as a set of one-dimensional modes (in the amount of 3N). In this case, the large partition function over the energy states of the background will be equal to
Zph,g (v, T) = Qph,
3N
+tt
£
j=0
exp
V j + \)
kB T
3N
exp
2kb T
1 — exp (— kbpT
3N
To obtain the total energy of thermal vibrations, it is necessary to average the free energy of phonons fphg (T, v) = —kBT ln (Zph,fl (v,T)) over the entire spectrum
Fph,g = fph,g(T,v )gph,g(v )dv, 0
where gph,g (v) is a density of phonon states. We choose the following normalization of
the density of states f0 gph,g(v)dv = 1. Then the total energy of thermal vibrations is equal to
3 n+tt
Fph,g = ^ Nohp vgph,fl (v )dv +3NokB T
V-0-V-' ^
h
C lK1—exp (—kBT
gph,g (v )dv.
Ig
hp v
We calculated the first integral I1 within the Debye approximation. For the second integral I2 it is worth noting that under the integral there is a special case of the polylogarithmic function ^Lifc(s) = ^, a special function, which is a
generalization of the Riemann zeta function. For s = 1 put the polylogarithm in the form Li i(s) = — ln(1 — s). We replace the polylogarithm with the sum of an infinite series. After that, we change the order of integration and summation and use the integral mean value theorem. The equation for the total energy of thermal vibrations can be expressed as follows:
Fph,g = 8NokBTD + 3NokBTln — exp ^—TSjT)
where hpVs(T) = Ts(T) is the mean temperature expressed in terms of the mean frequency resulting from the use of the mean theorem.
To take the thermal expansion of the crystal lattice into account, we introduce the anharmonicity of atomic vibrations into our consideration. We use the definition of the Griineisen parameter for the Debye model, taking into account that in the operating temperature range it does not depend on volume. By solving the following differential equation
Y(V) = — () = const > 0
V d ln(v) js= const
we obtain the dependence of the Debye temperature TD on the volume and assume a similar type of the dependence on volume for the temperature Ts,
Td (T,w) = Td,o = Td,oId (w), Ts(T,w) = Tsfl(T) = Ts,o(T )fa(w).
fD M
It is necessary to estimate the value Ts,o(T). This can be done by means of two different approaches: using the Debye model or using numerical integration of phonon density state functions. Below, we compare both approaches. We set equal the expression for the second integral I2 before and after using the mean value theorem in the framework of the first approach. At the next step, we substitute the density of phonon states formula from the Debye model and use the definition of the lower incomplete gamma function, letting the relative volume change tend to zero (w ^ 0), since the maximum value wmax = 0.011 for La(Fe,Si)13 is small. The obtained equation for the relative mean temperature is presented below:
Ts,o(T) T
TC,o TC,o
0s(T)
3ln( T^l — ln(3) — ln (r (3, 0, (Td,o/T)))
In the framework of the second approach, we calculated the parameter Ts,o(T) by numerically integrating the phonon density of states calculated by means of the DFT formalism in [21] using formula
Ts,o(T) T i f+~ f hpv
ln{i exK—kBk) gph'a(v)dv)
Tc,o Tc ,o \Jo V kB T
ds (T)
Fig. 1 depicts a comparison of the temperature dependences of the mean temperature, calculated by the two approaches. In the vicinity of the phase transition temperature,
7(K)
Fig. 1. Temperature dependences of the relative mean temperature calculated in the framework of the Debye model approximation and by numerically integrating of phonon density of state [21]
the value 0(s,O) = ds « 1.31, and the absolute value of the mean temperature is equal to T(s 0) « 257 K; thus, this value will be constant in subsequent calculations.
At the next step, we obtain the phonon entropy as the derivative with respect to temperature of the total energy of thermal vibrations:
Vg+a) = 3No|exp (I^ — 1] — — — exp (—Jfs^ } .
1.3. State equations system
According to the invariant of the first differential, one can pass from partial derivatives -JT, dH, dT to partial derivatives with respect to -Jr~, jt :
/ dF(T,p,H,^,m)
V / m,T,p,H / dF(T,p,H,u,m) \
V dm >L,T,p,H
= -P, = 0.
Bean and Rodbell model [18] Generalized model [this work]
Fig. 2. Volume dependence of the Curie temperature takes into account the root dependence of Tc
The classical Bean — Rodbell approach uses a linear approximation to describe the dependence of the phase transition temperature (Curie temperature) TC (p,u) = TC,0 (1 + fiu) on the relative volume change. In our model, we use a power dependence TC (p,u ) = TC0 (1 + u )p (p), where fi = '
07 ^ and pc is the critical
2ko Tq,o\/Pc -p ^c
pressure at which the Curie temperature becomes zero. This type of dependence on pressure described in the articles [20;
22; 23]. The use of such an approximation made it possible to significantly expand the region of the considered volumes, and as a consequence, the pressure span.
Summarizing all the obtained expressions for each contribution, the equation for the total Helmholtz free energy (1) of the system can be written as follows:
F(T,p,H,u,m) = -ß0HM0m - 2 (-J^j UoksTc,o(l + u)ß(p)m2-
'sinh (JB-1(m))\ I u2
-n°k*T < ln I Sinh (JJm)) ) - mB-1 (m)+ -
-3NokB< t-( / fS((Jl)-TT - l^1 - exP i-Vfs(u) - TSeh
[exp (ffs(u)) - 1] ^V t ^
Because we assume that Sel does not depend on relative volume change or magnetization, its specific form will be of no interest to us in subsequent discussions.
1.3.1. State equation for relative volume change
Consider the derivative of the Helmholtz free energy with respect to the relative volume change.
'dF(T,p,H,u,m)\ 1 ( 3J \ _ 2
-a;- = - HJ+rJnokBTcMj(1+u)" m +
/ m,T,P,H
i -T (dSPh,9+a(T,u)\ = -p.
ko \ou
\ / m,T,p,H
The derivative of the phonon entropy with respect to the relative volume change is presented in the following form.
' d^gM ^ 3NokBY (?SL \2 1 1
du 0 B ' V t J (1 + u)2y+y 4 sinh2 (§ fs(;)) '
v-v-'
Expand the factor $(t,u) in a series up to the first degree in relative volume change. We introduce the following substitutions: £ = 1 (j+y) n0kBk0TC0; x = pck0;
p = p/pc; a0 = 3N0kBk0Y; T (6f g2 = Tcflds (e~fg; «0^,0 = W 0S (6f g j-h^j =
6 s g 1
9s (Vf) —, A>s \—t = ^(t); ^0^(t) = <p(t). Having expressed from the resulting equation
2 |cosh ( -f )-1J
the relative volume change, we obtain the following state equation
£ß(p)m2 + p(t) - XP
f0(t) - (p)(P(p) - 1)m2' where f0(t) = 1 + p(t) [1 + 27 (1 - (|g coth (|gg].
1.3.2. State equation for magnetization
At this stage we consider the derivative of the Helmholtz free energy with respect to magnetization
'dF(T,p, H, u, m)
dm
' u,T,p,H
-ßoMoHoh-
3J
nokßTco(1+ w)ß2m — T
dSmag (m)
J +1/ 0 B c,ov ■ i dm
' u,T,p,H
To calculate the partial derivative of the magnetic entropy (3) with respect to the magnetization, it is necessary to know the general form of the inverse Brillouin function. In [24], the approximation of the inverse Brillouin function is presented in the following form:
i, N aJmJ2
. L ( im I rsj
J
B-1 (m)
1 — bm2'
where aJ = (1+JJ^;^-1—<2'?0)55) + J, b = 0.8. It was mentioned that such an approximation works well in the interval J G [1; 10] and |m| < 0.99. In Fig. 3 the numerical calculation of the inverse Brillouin function and the chosen approximation for the case J =1/2 are presented.
10-
cn
~ 5
CO
c\i
0Q
-5-
-10-
Numerical calculation © Approximation from [24]
G
-1,0
-0,5
0,0
0,5
1,0
M/Mq (arb. units)
Fig. 3. Comparison of the inverse Brillouin function obtained from the numerical calculation and using the approximation from [24]
Thus, Smag can be presented as follows
Smag nk
ß
ln
sinhi JW-
a j m2 J2 (1 — bm2)
sinh
Ja j m
2(1-bm2
In turn, the partial derivative with respect to magnetization is expressed as
dSmag (m ) dm
= Jaj (1 + bm2) (J sh ( J — J sh ( JajmJ1)) + sh (JJ
2(b*m — 1)2 sh (J) sh ( JJ+U
The final state equation for magnetization: ^o Mo Ho ( 3J
2J 2 aW m (bm2 — 1)2
h-
no kß Tc,o
J +1
^ (1 + ^)ß m + t
dsSmag (m) dm
w,T,p,H
Expansion at w^o 1+ßw
Taking a combination of constants on the left side of the equation
_Mo
no kB Tc
kBTc,° = i rewrite the equation in the form
,0 1 M0 MB ' ^
-j Tr\-MM Ho
n° kb Tc,° o o
Mo Ho =
h = t
dSmag (m) dm
u>,T,p,H
_n, 2 (P)m2 + 0 (p)<p(t) - P (p)xp\ m 1 + fo(t) - ep(p)(P(p) - i)m2 ; •
(4)
2 (T)
Fig. 4. Calculated (solid lines) and experimental (symbolic plots) [20] field dependences of magnetization for LaFen.4Sii.6 compound
2. Numerical simulation
In this section, the verification of the modernized phenomenological model of the first-order metamagnetic phase transition for La(Fe,Si)13 compounds underwent was carried out. One of the undeniable advantages of our model is its ability to predict temperature and field hysteresis.
The magnetization field dependences were calculated using the following algorithm. The h(m) dependences were plotted using (4) in the first stage. The metastable state corresponds to the section of the obtained dependence with a negative slope (jm < 0). This section was excluded by connecting the points of maximum and minimum on this graph (dm = 0). Fig. 4 depicts calculated (solid lines) and experimental (symbolic plots) field dependences of magnetization for the LaFen.4 Si1.6 compound under ambient pressure. The dashed line corresponds to the aforementioned procedure of plotting. According to these graphs, it can be seen that the developed model correctly describes the critical magnetic fields at which a transition from the paramagnetic to the ferromagnetic state is induced.
The temperature dependences of magnetization under various external magnetic fields are shown in Fig. 5. Dependence m(T) was built using the set of m(h) dependences, calculated for a wide range of temperatures, by taking the values of a constant field. The shape of the magnetization temperature dependences qualitatively agrees with the experimental results [20].
Fig. 5. Calculated (solid lines) and experimental (symbolic plots) [20] temperature dependences of magnetization for LaFe11.4Si1.6 compound
The analysis of the obtained simulation results points out that the estimated upper limits of the field and temperature hysteresis are overestimated. The main reason is that kinetic is not taken into account. In [25; 26], the relaxation dependences of magnetization were measured, which revealed that the equilibrium magnetization state occurs in hundreds of seconds. In [27], some attempts were made to incorporate the kinetics into the thermodynamic model. The authors combined the phenomenological model of Kolmogorov — Johnson — Mehl — Avrami with the Bean and Rodbell model for the description of the time-dependent evolution of phase transformation in La(Fe,Si)13 compounds. Their approach led to a reduction of the hysteresis value by several times.
The developed generalized phenomenological model made it possible to estimate the required pressure value in order to achieve an almost hysteresis-free magnetization-demagnetization cycle. According to the results of the numerical simulation of the multi-stimuli cycle presented in Fig. 6, it was found that it is necessary to magnetize under ambient pressure, and demagnetize at p = 39 MP a. These values coincide in order of magnitude with those found experimentally in the article [7].
Fig. 6. Field dependences of magnetization calculated at zero pressure and 39 MPa
Conclusions
In this work, the generalized phenomenological model of the first-order metamagnetic phase transition was developed: the nonlinear dependence of the exchange energy on the relative volume change was taken into account, the contribution of phonon entropy to the Helmholtz free energy of the system was revised, considering the harmonic and anharmonic approximations, the magnetic entropy contribution was determined using the inverse Brillouin function.
Our model describes with sufficient accuracy the peculiarities of a first-order phase transition under the simultaneous action of several generalized forces: temperature, magnetic field, pressure. This made it possible to elaborate the upper limit of the field and temperature hysteresis widths, as well as to estimate the external pressure values necessary to obtain a hysteresis-free magnetization-demagnetization cycle.
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Article received 09.02.2023.
Corrections received 10.06.2023.
Челябинский физико-математический журнал. 2023. Т. 8, вып. 2. С. 280-291.
УДК 537.638.5 Б01: 10.47475/2500-0101-2023-18211
ОБОБЩЁННАЯ МОДЕЛЬ МАГНИТОСТРУКТУРНОГО ФАЗОВОГО ПЕРЕХОДА В СПЛАВАХ La(Fe,Si)lз ПРИ ОДНОВРЕМЕННОМ ВОЗДЕЙСТВИИ ТЕМПЕРАТУРЫ, МАГНИТНОГО ПОЛЯ И ДАВЛЕНИЯ
Р. А. Макарьин1'", М. В. Железный2'3 6, Д. Ю. Карпенков12с
1 Московский государственный университет им. М. В. Ломоносова, Москва, Россия 2Национальный исследовательский технологический университет «МИСиС», Москва, Россия
3Институт металлургии и материаловедения РАН им. А. А. Байкова, Москва, Россия "makarin.ra16@physics.msu.ru, 6markiron@mail.ru, сkarpenkov.dy@misis.ru
Представлена модель метамагнитного фазового перехода первого рода для сплавов типа Ьа(Ее,8^1з в приближении локализованных моментов в присутствии нескольких обобщённых сил. Для достижения максимальной эффективности устройств магнитного твердотельного охлаждения необходима точная настройка температуры Кюри внешними воздействиями (магнитным полем и давлением). Представленные в литературе на сегодняшний день термодинамические феноменологические модели фокусируют своё внимание на описании магнитокалорического эффекта, почти не уделяя внимания барокалорическому эффекту, хотя он, в свою очередь, способен обеспечить значительное уменьшение паразитного эффекта гистерезиса в цикле намагничивание-размагничивание, а также сместить температуру Кюри вплоть до криогенных температур. Всё вышеперечисленное требует не только ухода от приближения линейной зависимости температуры фазового перехода от изменения объёма, но также модернизации вида магнитной и фононной энтропии с учётом ангармонических поправок. В данной работе был расширен рабочий диапазон давлений, получены уравнения состояния для магнетика при фазовом переходе. Построенная в данной работе модель позволила провести оценки верхнего предела температурного
Исследование выполнено при финансовой поддержке Российского научного фонда, грант № 21-72-10147, и Министерства науки и высшего образования РФ, грант № 075-15-2021-1353.
и полевого гистерезиса и предсказать необходимую величину внешнего давления для уменьшения паразитного влияния гистерезиса.
Ключевые слова: магнитокалорический эффект, функция Бриллюэна, фононная энтропия, феноменологическая модель, метамагнитный фазовый переход.
Поступила в 'редакцию 09.02.2023. После переработки 10.06.2023.
Информация об авторах
Макарьин Родион Алексеевич, аспирант, младший научный сотрудник кафедры магнетизма, Московский государственный университет имени М. В. Ломоносова, Москва, Россия; e-mail: makarin.ra16@physics.msu.ru.
Железный Марк Владимирович, научный сотрудник кафедры физического материаловедения, Национальный исследовательский технологический университет «МИСиС», Москва, Россия; научный сотрудник, Институт металлургии и материаловедения РАН им. А. А. Байкова, Москва, Россия; e-mail: markiron@mail.ru.
Карпенков Дмитрий Юрьевич, кандидат физико-математических наук, доцент кафедры функциональных наносистем и высокотемпературных материалов, Институт металлургии и материаловедения РАН им. А. А. Байкова, Москва, Россия; старший научный сотрудник кафедры магнетизма, Московский государственный университет имени М. В. Ломоносова, Москва, Россия; e-mail: Karpenkov.dy@misis.ru.
