Statistical model for the martensitic transformation simulation in Heusler alloys

Journal Pre-proof Statistical model for the martensitic transformation simulation in Heusler alloys Olga N. Miroshkina, Vladimir V. Sokolovskiy, Danil R. Baigutlin, Mikhail A. Zagrebin, Sergey V. Taskaev, Vasiliy D. Buchelnikov

PII: DOI: Reference:

S0921-4526(19)30756-2 https://doi.org/10.1016/j.physb.2019.411874 PHYSB 411874

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Physica B: Physics of Condensed Matter

Received date : 28 June 2019 Revised date : 27 September 2019 Accepted date : 6 November 2019 Please cite this article as: O.N. Miroshkina, V.V. Sokolovskiy, D.R. Baigutlin et al., Statistical model for the martensitic transformation simulation in Heusler alloys, Physica B: Physics of Condensed Matter (2019), doi: https://doi.org/10.1016/j.physb.2019.411874. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

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Statistical model for the martensitic transformation simulation in Heusler alloys Olga N. Miroshkina∗, Vladimir V. Sokolovskiy, Danil R. Baigutlin, Mikhail A. Zagrebin, Sergey V. Taskaev, Vasiliy D. Buchelnikov

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Chelyabinsk State University, 454001 Chelyabinsk, Russia

Abstract

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In this work, the one-dimensional statistical model based on the phenomenological approach for the study of the magnetic and magnetocaloric properties in Heusler alloys is considered. The proposed model is approved for Ni-Co-MnIn exhibiting both the first-ordered structural transformation and second-order

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magnetic phase transition. Temperature dependencies of magnetization and magnetic entropy change under applied magnetic field and external pressure are studied. The results of the simulations are in good agreement with the experimental data.

Keywords: multifunctional materials, magnetocaloric effect, statistical modeling, phase transitions

1. Introduction

The energy-efficient technologies have attracted significant attention from the scientific community in recent years. One of the key technology is the magnetic refrigeration at room temperatures that considered as an environmen-

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tally friendly, high-efficient, and green cooling with respect to the conventional gaseous refrigeration. The magnetic cooling is based on the magnetocaloric effect (MCE) that describes as the thermal response of magnetic material on the ∗ Corresponding

author Email address: [email protected] (Olga N. Miroshkina)

Preprint submitted to Physica B: Condensed Matter

September 27, 2019

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application of the external magnetic field under adiabatic or isothermal condi-

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tions. To date, a wide range of magnetic materials exhibiting MCE has been considered as the promising refrigerants [1, 2, 3, 4, 5, 6, 7, 8]. Among these systems, Ni-Mn-based Heusler have attracted much attention due to their multifunctional

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properties such as the shape memory effect, magnetic-field strains, large magnetocaloric, barocaloric, and magnetoresistanse effects [3, 4, 8, 9, 10, 11, 12, 13, 14, 15]. These properties are associated with the structural phase transition between the high-temperature austenitic phase and the low-temperature martensitic one. Unfortunately, comprehensive use of Heusler alloys for the magnetic

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refrigeration systems is hampered by thermal hysteresis that takes place across the martensitic transformation. The presence of hysteresis results in the degradation of the magnetocaloric properties after magnetization/demagnetization cycles due to the structural transformation processes [16, 17]. Thus, the ma-

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nipulation of the hysteresis width is the basic aspect of driving the reversibility of MCE [18]. Related to this problem, computer modeling is especially helpful and it allows to simulate the compound properties under different conditions without time-demanding, complex, and expensive experiments. In this paper, to develop the model for magnetic and magnetocaloric properties in Heusler alloys exhibiting martensitic transformation, we consider the phenomenological approach based on Malygin theory of the smeared phase transitions [19], Bean-Rodbell theory of the first-order phase transitions [20], and the mean-field theory [21]. Validation of our model is performed for nonstoichiometric Ni1.83 Mn1.46 In0.54 Co0.17 , which is potentially applicable as refrigerant. This compound exhibits separate first-order structural transformation

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and second-order magnetic phase transition [22].

2. Theoretical model Our model takes into account the coexistence of martensitic (m) and

austenitic (a) structural domains in the vicinity of the structural phase tran-

sition. The volume fractions of the structural domains ξα = Vα /V (Vα is the

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volume of α phase, V is the total volume) are calculated using the theory of "

∆V Qρ 1+exp kB Tm

T − Tm ± ∆Tm σEb ∆M µ0 H − − Tm Qρ Qρ

!#)−1

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ξm =

(

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diffuse martensitic transitions proposed by Malygin [19]

, ξa = 1−ξm , (1)

where ∆V is the elementary transformation volume, Q is the latent heat of martensitic transition, ρ is density of the compound, kB is the Boltzmann constant, Tm and ∆Tm are the martensitic transformation temperature and halfwidth of hysteresis loop at structural transition, σ is the external stress, Eb

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is the Bain strain denoted the transformation strain between martensitic and austenitic variants, ∆M is the change of magnetization at structural transition, and µ0 H is the external magnetic field. We would like to note that the ∆Tm term allows us to change the hysteresis width. According to Malygin et al. [19], this term is associated with the concentration of point defects and defined as 2 3kB Tm πQr 3 Cdef ,

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∆Tm =

where Cdef is the defect concentration, r is the defect ra-

dius. Both the smearing of martensitic transition and appearance of hysteresis are caused by an interaction between martensite/austenite interphase boundary and various defects in the real crystal.

The density of the total free energy includes austenite and martensite contributions consisting of the elastic (el), magnetic (mag), and magnetoelastic (me) terms

F = ξm Fm + ξa Fa − σE,

(2)

where

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Fα = Fαel + Fαmag + Fαme .

The elastic part of free energy (Fαel ) is obtained by integrating the relations

between the free-energy per unit mass and the elastic modulus C, thermal expansion coefficient ς, and specific heat c [23]. In contrast to Landau theory for

cubic ferromagnets [24], in which the temperature-dependent shear modulus C 0 is assumed, we consider here that C, ς, and c are temperature-independent and

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equal for austenitic and martensitic phases. For the present model, the elastic

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modulus C corresponds to the bulk one and is obtained as a second derivative of the free energy. It is well known from the experimental findings that bulk modulus decreases almost twice near melting point, while it is almost constant

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for lower temperatures, at which martensitic transition occurs.

The magnetic part of the free energy Fαmag is defined in the framework of the mean-field theory, while the magnetoelastic part (Fαme ) is taken from BeanRodbell theory [20]. Discussed above terms take the form

el Fm

 A y2  Rρ α + µ0 HM0α y + T Sαmag , 2 µ

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Fαmag = −

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! CE 2 T = − ςCE(T − Tm ) + ρcT 1 − ln , 2 Tm ! ! T T C(E + Eb )2 − ςC(E + Eb )(T − Tm ) + ρcT 1 − ln −Q 1− , = 2 Tm Tm Fael

Fame =

Ba y 2 E Rρ me Bm y 2 (E + Eb ) Rρ , Fm = . 2 µ 2 µ

Here Aα is the exchange constant, M0α is the magnetization saturation, y is the normalized magnetization, R is the universal gas constant, µ is the molar mass, Sαmag is the magnetic entropy, and Bα is the magnetostriction constant. The exchange constant and saturation magnetization are determined as Aα =

3Jα TCα µB gα Jα , M0α = , Jα + 1 kB

where Jα is the total angular momentum, TCα is the Curie temperature, gα is the Land´e factor, and µB is the Bohr magneton.

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The magnetostriction constant Bα is defined as s 18CTCα ηα ((2Jα + 1)4 − 1) µ . Bα = 80(Jα + 1)4 Rρ

Here the parameter ηα determines the order of the phase transition [20, 21]. To express the strain and magnetic order parameters, it is necessary to

minimize the free energy function (2) with respect firstly to E and then to y. 4

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Thereby, the equation for normalized magnetization that takes into account the

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Brillouin function BJ can be written as " # 2Jα + 1 (2Jα + 1)Yα 1 yα = BJ Y α = coth − coth 2Jα 2Jα 2Jα

Yα =

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where

! Yα , 2Jα

i σBα  Rρ Bα2 yα3 1 h Aα + ςBα (T − Tm ) + yα + + M0α µ0 H T C µ 2C

Thus, the total magnetization of the system =

M

=

R H  H coer M0a ξa ya coer + M0m ξm ym coer for H < Hall , µ Ha Hm R coer (M0a ξa ya + M0m ξm ym ) for H > Hall , µ

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M

(3)

coer coer where Hall = Hacoer + Hm is the coercive field strength of the whole com-

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pound.

Having calculated magnetic order parameter yα , the strain E can be determined as follows E=

2 Rρ (Ba ξa ya2 + Bm ξm ym ) σ + ς(T − Tm ) + ξm Eb + . µ 2C C

Finally, the magnetic part of entropy can be written in accordance to meanfield theory

mag S mag = Samag ξa + Sm ξm ,

where

Sαmag

"

R = ln µ

sinh



2Jα +1 2Jα Yα

sinh



Yα 2Jα





#

− yα Yα .

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To calculate the isothermal entropy with varying external magnetic field, we make use of

∆S mag (T, µ0 H) = S mag (T, µ0 H) − S mag (T, 0).

Here, S mag (T, µ0 H) and S mag (T, 0) denote the entropy in presence of a

magnetic field µ0 H and in zero field, respectively. 5

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3. Calculation details

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As was mentioned above, the verification of considered model is performed for the non-stoichiometric Ni1.83 Mn1.46 In0.54 Co0.17 , which exhibits record values

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of inverse MCE. The comprehensive experimental findings for this composition are presented in Ref. [25].

The structural subsystem is defined by the following parameters. The molar mass µ is 0.26 kg/mol. The density of alloy ρ ≈ 7900 kg/m3 is evaluated as a ration of molar mass to molar volume. In the absence of the external pressure, the structural transition temperature Tm = (As + Af + Ms + Mf )/4 and hysteresis

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half-width ∆Tm are estimated from experiment [22] and taken 280 K and 8 K, respectively. Here, As , Af , Ms , and Mf are temperatures of austenite start, austenite finish, martensite start, and martensite finish, respectively. While for pressures σ = 0.45 and 0.84 GPa, the values of ∆Tm are 7 K and 6.5 K. The

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elementary transformation volume ∆V is 2×10−26 m3 . Experimental study of Ni-Co-based Heusler alloys [26] has shown that the value of latent heat Q depends strongly on composition and decreases from 8000 to 2000 J/kg with Co doping. Therefore, the average value of Q = 5000 J/kg is used for modeling. The first-principles calculations for similar composition yields equal bulk moduli for austenite and martensite [28]. Thus, for our calculations, C = 135 GPa is taken into account.

The magnetic properties are specified by the parameters as follows. The Curie temperatures of austenite TCa = 390 K and martensite TCm = 150 K as well as the change of magnetization at structural transition ∆M = 90 Am2 kg−1 are evaluated from the experiment [22]. To estimate a saturation magnetiza-

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tion M0α via a total angular momentum Jα , the total magnetic moment of 6.6 µB /f.u. for austenite and 1.2 µB /f.u. for martensite are used from ab initio calculations [29]. Thus, Ja and Jm are taken as 3.3 and 0.6, while the Land´e

factor ga,m is equal to 2. Such a large difference between magnetizations is accounted for the ferrimagnetic ordering of the martensitic phase. According to the experimental studies [27], coercive field strength is Hacoer = 0.015 T for

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coer austenite and Hm = 0.04 T for martensite. Concerning the proposed model,

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a coercive field influences on the magnetization in the case of a low magnetic coer field (H < Hall ). However, our modeling was performed for magnetic fields coer H > Hall , consequently, the values of coercive forces are used only as a condi-

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tion for choice of the equation for magnetization (Eq. 3).

4. Results and discussion

Temperature dependencies of magnetization in the external magnetic field 1 T calculated for heating and cooling protocols at different external pressure

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are presented in Fig. 1. It is seen the applying of the external pressure shifts the structural transition towards higher temperatures. This implies martensitic structure can exist in a wider temperature range. The magnetic transition temperature of the austenite increases in the presence of the pressure but this

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shift dTC /dσ is smaller than dTm /dσ for structural transition temperature. The magnitude of dTm /dσ shift is in accordance with the experiment [22], as shown in Fig. 1. However, there is no experimental evidence of dTC /dσ in magnetic field 1 T for Ni-Co-Mn-In. Nevertheless, according to Kanomata et al. [30], the shift of the Curie temperature might be variated from +0.9 K/kbar for Ni2 MnIn to +4.1 K/kbar for Ni2 MnSb in low magnetic fields (50 Oe). Our value is dTC /dσ = 3.57 K/kbar in magnetic field 1 T that belongs to the pointed range.

The set of isofield thermomagnetization curves of Ni1.83 Mn1.46 In0.54 Co0.17 across martensitic transformation upon heating and cooling is depicted in Fig. 2. It is seen from this figure that for both 0 and 0.84 GPa, the martensitic tran-

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sition temperature is found to be reduced with an increasing magnetic field. This behavior shows clearly that an austenite phase is stabilized by an applied magnetic field. In addition, having found the martensitic transition temperatures (As , Af and Ms , Mf in Fig. 2), next we analyze the (Tm -µ0 H) curves as illustrated in Fig. 3. The

1 dTm µ0 dH

curves are found to be slightly decreased

with field enhancement from 0.1 to 2 T. The applying of the external pressure

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Figure 1: Magnetization as a function of temperature for Ni1.83 Mn1.46 In0.54 Co0.17 at pressure 0, 0.45, and 0.84 GPa. The magnetic field is 1 T. Open symbols denote the experimental data upon heating and cooling protocols [22].

Figure 2: Magnetization as a function of temperature for Ni1.83 Mn1.46 In0.54 Co0.17 at pressure

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(a) 0 GPa and (b) 0.84 GPa for the different magnetic field.

results in a decrease in

1 dTm µ0 dH

slope reproducing qualitatively the experimental

behavior [22]. By applying and varying stress, one can change the temperature range of austenite-martensite co-existence in a magnetic field and achieve MCE at room temperatures. However, the change of slope is not so notable 8

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in comparison with the experiment [31, 22]. It can be related to the fact that

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in our case the applied stress shifts the transition temperature less than it is observed experimentally (see Fig. 1). This discrepancy can be accountable for the following reasons. The model considered takes into account one-dimensional

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settings, which corresponds to uniaxial stress in a crystal, while experimental measurements for Ni1.83 Mn1.46 In0.54 Co0.17 have been performed under hydrostatic pressure. Nevertheless, it gives an adequate qualitative description of the

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magnetic properties of the crystal near the phase transition.

Figure 3: The field dependencies of the temperatures austenite start As , austenite finish Af , martensite start Ms , and martensite finish Mf for Ni1.83 Mn1.46 In0.54 Co0.17 at pressure (a) 0 GPa and (b) 0.84 GPa. The lines denote a linear approximation.

Fig. 4 displays the temperature-field dependences of magnetic entropy for composition studied in the vicinity of martensitic transformation at an applied pressure of 0.84 GPa. It is seen that the magnetic entropy shows the jump-like hysteresis behavior across the structural transition between the low-magnetic martensitic phase and the ferromagnetic austenitic phase. Moreover, an increase in the magnetic field up to 5 T results in a linear reduction of magnetic entropy mag

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( µ10 dSdH

≈ 0.26 J/(kgKT)) due to increasing of magnetic order in austenite.

Besides, the martensitic transformation region is shifted toward room temperatures by applying the magnetic field. The results of MCE modeling are presented in Fig. 5. Fig. 5(a) shows the in-

verse MCE (∆S mag ) observed at the martensitic transformation in comparison

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Figure 4:

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Three-dimensional temperature-field dependence of the magnetic entropy of

Ni1.83 Mn1.46 In0.54 Co0.17 upon heating and cooling at the pressure of 0.84 GPa.

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with experimental values at magnetic field changes of 1 and 2 T under pressure. The experimental values [22] have been obtained from the heat capacity measurements. The enhancement of the magnetic field from 1 to 2 T leads to much strongly pronounced MCE that increased by more than 1.33 times. Meanwhile, with gain in external pressure from 0 to 0.84 GPa, the picks of MCE shift from 285 to 330 K and the MCE magnitude rises. This enhancement is caused by a decrease of magnetization under pressure. The curves modeled are in good agreement with the experiment.

The calculated both inverse (∆S mag > 0) and direct (∆S mag < 0) MCE upon magnetic field variation from 0 to 2 T are depicted in Fig. 5(b). Notice, the direct MCE is observed at ferro-paramagnetic phase transition in the austenitic phase. This transition is of second-order one without thermal hysteresis as

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compared to the martensitic transformation. As a consequence of this, the values of direct MCE are smaller sufficiently in comparison to those for the inverse MCE, where both magnetic and structural subsystems play a role. As can be seen from this figure, the application of the external pressure shifts both inverse and direct MCE towards high temperatures. Moreover, the magnitude of inverse MCE is found to increase with increasing pressure while the value of 10

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direct MCE changes insignificantly.

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Figure 5: Magnetic entropy change as function of temperature for Ni1.83 Mn1.46 In0.54 Co0.17 at pressure 0, 0.45, and 0.84 GPa. (a) The calculated inverse MCE in comparison with the experiment [22] at the magnetic field change of 1 and 2 T for the heating protocol. (b) The calculated inverse and direct MCE at the magnetic field change of 2 T for heating

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and cooling protocols.

5. Conclusion

In this paper, the statistical model based on Malygin theory of the smeared phase transitions, Bean-Rodbell theory of the first-order phase transitions, and the mean-field theory is proposed for magnetic and magnetocaloric properties of Heusler alloys. Non-stoichiometric composition Ni1.83 Mn1.46 In0.54 Co0.17 has been considered as an example. It is shown that the manipulation of the martensitic transformation temperature can be realized by applying the external pressure and magnetic field to a compound instead of a variation in chemical composition. In contrast to the magnetic field, the applying of pressure leads to an

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increase in structural transition temperatures. The value of MCE enhances with increasing both the magnetic and uniaxial stress fields. The results of modeling are in good agreement with the available experimental data.

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Acknowledgments

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This work was supported by the Russian Science Foundation No. 17-72-

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20022.

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Conflict of Interest and Authorship Conformation Form Please check the following as appropriate:

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o (x) All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. (x) This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.

o

(x) The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

o

The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript: Affiliation Chelyabinsk State University Chelyabinsk State University Chelyabinsk State University Chelyabinsk State University Chelyabinsk State University Chelyabinsk State University

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Author’s name Olga Miroshkina Vladimir Sokolovskiy Danil Baigutlin Mikhail Zagrebin Sergey Taskaev Vasiliy Buchelnikov

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o