MIXING ENERGY OF Fe-Cr ALLOYS: FIRST PRINCIPAL

CALCULATIONS

A.A. Mirzoev , M.M. Yalalov , D.A. Mirzaev

Southern Ural state university, Chelyabinsk, Russia

In this work dependencies of the average magnetic moments on atom and mixing energy of the alloy from chromium concentration in a substitutional solid solution of iron are considered. An attempt to explain abnormal behaviour of characteristics of alloys (near order, Curie point, electroresistance, lattice parameter) at very small chromium concentration (less than 10 at.%) is made. Occurrence of chromium atom causes disturbance of magnetic structure in an extensive zone of a α-iron lattice, that surrounds impurity atom, forming a cluster, strongly distinguished on properties from pure substance. At increase of concentration, zones of magnetic disturbance in the iron lattice start to overlap, thus leading to reduction of chromium-iron interaction energy when хCr > 0.3. At the further growth of chromium concentration, parameters of interaction depend on concentration a little

1. INTRODUCTION

System Fe-Cr plays an exclusive part at designing special steels [1]. On reaching in the iron alloy 13at.% of chromium electrochemical potential rises stepwise in the area of positive values. Thus alloys and steels with higher concentration of chromium turn out to be stainless regardless of the phase composition. Alloys, containing more than 13 at.% of chromium, are ferritic under all temperatures, up to melting. They have body-centred cubic lattice and possess a ferromagnetism below Curie temperature (for compositions with хCr < 0,8) or antiferromagnetism below Neel temperature (for хCr > 0,8). Heating and cooling of such steels are conditioned on necessity of a recrystallization annealing after rolling. At the analysis of the annealing is cleared up that decelerated cooling or exposure of products at 450-485ºС cause their most strong embrittlement, in connection with this appear a term "brittleness 475".

It was offered many hypothesises about the nature of this type of brittleness. For instance, idea was offered about the evolution in γ-phase of processes of sequencing, which stimulate hardering and embrittlement [2]. However thermochemists help to decide this problem. Measurements of heat of formation the alloys Fe-Cr from β-iron (α-iron at temperatures above Curie point) and chromium [3] gave positive values. This means that unlimited solutions are stabilized by the configurational entropy at high temperatures. In case of its reducing any solid solution of this systems must stratificate on two solutions – enriched and depleted by chromium, but with same bcc lattice. Repeat examination of heat of formation [4] gave same result, moreover numerical agreement of data was fine. It turned out that Fe-Cr solutions are closely to regular ones. All whole of data about behaviour of solid solutions of iron-chromium system characterize its as a typical system with positive heat of formation, and therefore it is inclined to stratification.

However in last years have appeared researches [5] of near order in the iron-chromium system, in which conclusion was made that at relatively small chromium concentration near order corresponds to an ordering, but at high concentrations it corresponds to stratification. These results contradict with foregoing features of system and, certainly, need improve of the physical nature of ordering anomaly. At present, due to the appearance of high-performance computers, a possibility has appeared to calculate disordered substitutional alloys by the supercell method, which was developed for metallic glasses investigation [6]. In this method selfconsistent zone calculation of electronic structure is adjasted to the fictitious crystal, formed by the periodically reiterative in the space elementary cell, as which a certain portion of investigation disordered substance is taken. Necessity of calculation of the cell with the many atoms has conditioned using of program package TB-LMTO, which was developed by the Ole Кrogh Anderson group [7]. At present a linear "muffin-tin" orbitals (LMTO) is most fast and economical method of electronic structure calculation with the minimum basis: nine basis functions for s-, p-, d- states per atom. Its efficiency and accuracy are checked on multiple objects (for instance, [8]). Its essential advantage is a possibility of examination of partial density of states depending on spin for each orbital of each basis atom. Knowledge of these densities defines a possibility of study of magnetic properties of our system. Executed calculation has confirmed a presence of anomalies of average magnetic moment of iron atoms at small chromium concentrations and has allowed to clear reasons of this phenomena.

2. COMPUTITIONAL DETAILS

Calculation was made by means of the package TB-LMTO-ASA version 4.7. For modeling of different concentration values a supercell with 16 iron atoms in the right bcc lattice was choose; some of them were replaced by chromium atoms. Thus was performed modeling of alloys with the chromium concentration values from 0 to 100 at.% with the step 6.25 at.% (altogether 17 points). Small values of chromium concentration unachievable in the cell with 16 atoms (1.85 at.% and 3.7 at.%) were calculated in the cell with 54 atoms. Calculations of total energy, density of states and magnetic moments per atom for non-magnetic, ferromagnetic and antiferromagnetic cases were performed in approximation of local spin density with Barth-Hedin exchange-correlation potential [9]. The error connected with using of atomic spheres approximation (ASA), was corrected by means of so named combined corrections [7,10]. In all cases was used the LMTO-basis, included s-, p- and d- functions for each atom of supercell. At integration over the Brilluen zone was used mesh in -space with cells that provided in checking tests for accuracy of total energy calculation to within 0.5 mRy. We investigated concentration dependence of total energy and mixing energy of alloy. Mixing energy are found by subtraction of full energy of components, taken with appropriate weights, from the total energy of alloy:

, (1)

where Etot – total energy of Fe1-xCrx alloy, EFeиECr – total energies of corresponding chemical elements (all per atom).

3. RESULTS

Results of calculation of concentration dependence of the magnetic moments of iron and chromium atoms , and also the average moment on atom of an alloy are given on Fig. 1 in comparison with known literary data [12,13,14]. It is visible, that the magnetic moments per atom of each type and average magnetic moment per alloy atom are in good agreement with experimental data [11]. Growth of the average magnetic moment per iron atom is visible at the doping of the pure iron with small quantity of chromium (approximately 10 at.%) – this fact adjust well with the experiment, as well as that at approximately 20-40 at.% abnormal behaviour is changed into theoretically predicted decrease.

Fig.1 The calculated (lines with empty symbols) and measured (full symbols) magnetic moments versus Cr concentration: 1,4,7,10-average magnetic moment of alloy, 3,5,8 - average magnetic moment of Fe atoms, 2,6,9 - average magnetic moment of Cr atoms. (5,6,7-Aldred at al.data [11], 8,9-Kajzar at al. data [12], 10- thermodynamics calculation data[13])

Fig. 2.Chromium concentration dependence of mixing energy Fe-Cr system. – our LMTO results [ ] ;  - results of thermodynamic calculation [3].  – experimental data [4],.

The most interesting are results of calculations at low chromium concentrations: at the region 2-6 at.% of chromium a mixing energy turned out to be negative with the minimum near 3 at.% of chromium. It is possible to use these facts for the explanation of Sagaradze work [5], where it was observed near ordering in low-chromium alloys. Also agrees himself region of concentration values, where according to our data Есм0 and in [5] near ordering was observed.

Analysis of literary data about Fe-Cr system physical properties has allowed to reveal some features of their change at low chromium concentrations. For instance, concentration dependencies of electroresistance and lattice parameter [14] have an initial abrupt raising area (before 10 at.% of chromium), then rate of growth decreases and hereinafter stays permanent. A change of Curie point is worthy of special attention. Results of systematic measurements, executed by Adcock, in detail are provided in [15]. Chromium additions up to three atomic percent raise a Curie point from 769 to 780ºC, whereupon its reduction begins. Alloy with 7 at.% of chromium has such Curie point, as the pure iron. In the article [16] there are data about diffusion constants near Curie points. For the alloy with 3,09 at.% of chromium is given Curie point 777ºC, i.e. increasing by this data is 8 degrees. It is necessary to remember that Curie point characterizes energy of ferromagnetic interaction, and, according to calorimetric measurements, . Therefore it is interesting to testify the data to determined the perturbation of a magnetic subsystem of iron atoms caused by chromium impurity . For an estimation of this perturbation it is used the elementary average field approach:

, (2)

where defines ferromagnetic constituent of mixing energy, and Аij – exchange integrals. Values of this integrals , , provided an agreement of experimental and calculated points in the area xCr > 0.1. However it is impossible to describe the increasing of Curie point at low chromium concentrations by such values of parameters Аij: one has to enlarge greatly and to decrease . For instance, if at 3 at.% of chromium increasing of Curie point is 10 degrees, then the value of parameters should take equal: , . It follows that in the region of lowl concentrations magnetic constituent of mixing energy is , whereas at xCr > 0.1 magnetic constituent of mixing energy is about . Thermodynamic data at high temperatures, which is equal to , relates to non-magnetic mixing energy. It is obviously that their sum turns out to be negative (-15000) at low х and positive (+3200) at high х. Thence it is possible that negative sign of mixing energy is explained by features of chromium atoms magnetic state in the iron lattice at low concentrations.

Calculations of magnetic moments of iron atoms, surrounded single chromium atom, has revealed following regularities: nearest iron atoms have the magnetic moment less, than pure iron (2.229μB) – 2,183μB; on second and third neighbors increasing of magnetic moment is observed (2,278μB and 2,315μB accordingly); atoms in fourth, fifth and sixth neighbor sphere of chromium atom has magnetic moments less, than second and third neighbors, but greater, than pure iron (near 2,26μB); magnetic moment of seventh neighbors much more (2,437μB). Thereby, appearance of chromium atom causes a disturbance of magnetic structure in an extensive zone of a iron lattice, that surrounds impurity atom. It is probably that therefore energy of the magnetic (antiferromagnetic) interaction of the iron and single chromium atom turns out to be high and this leads to Curie point raising.

However increasing of the chromium concentration results in the appearance of second atom of chromium at first in distant, but then in more and more close neighborhood of the chromium atom. Since both atoms are antiparallel to the direction of magnetization, they turn out to be in the ferromagnetic interaction with each other. In one of the calculations attempt was made to obtrude on the second chromium atom an opposite direction of magnetic moment and state the value of its influence upon the total energy of alloy, when they are located nearby (0.866 lattice parameter) and when they are distanced (1.732 lattice parameter). And in that, and in other case a higher energy corresponded to the antiparallel directions of magnetic moments. Hence exactly iron lattice obtrudes on chromium the magnetic orientation. In Heisenberg model chromium atoms must be repulsed. It is probably that repulsing is intensified because of the overlapping of magnetic disturbances zones in iron lattice that provides in region хCr > 0.3 to reduction of the chromium-iron interaction energy and the Curie point. During the further growing of chromium concentration, parameters of interaction depend from it a little. Abnormal change of the mixing energy in the iron-chromium system is stipulated by strong disturbance of α-iron magnetic structure at very small chromium concentrations.

5. CONCLUSION

In this work dependencies of the average magnetic moments on atom and mixing energy of the alloy from chromium concentration in a substitutional solid solution of iron are considered. An attempt to explain abnormal behavior of characteristics of alloys (near order, Curie point, electroresistance, lattice parameter) at very small chromium concentration (less than 10 at.%) is made. Occurrence of chromium atom causes disturbance of magnetic structure in an extensive zone of a α-iron lattice, that surrounds impurity atom, forming a cluster, strongly distinguished on properties from pure substance. At increase of concentration, zones of magnetic disturbance in the iron lattice start to overlap, thus leading to reduction of chromium-iron interaction energy when хCr > 0.3. At the further growth of chromium concentration, parameters of interaction depend on concentration a little.

This work was supported by Russian grant of “MMC” and “Ausferr” Fund.

REFERENCES

  1. Goldstein M.I., Gratchev S.V., Veksler Yu.G.: Special steels. Moscow, MISIS,1999. In Russian.
  2. Bogachev I.N., Papina N.V.: ‘About nature of «475-degrees» brittleness of high-chromium steels’. Physical metallurgy and metals thermal treatment 1971 (5) 57-59. In Russian.
  3. Dench W.A.: ‘Adiabatic High Temperature Calorimeter for Measurement of Heats Alloying’ Trans. Faraday Soc. 1963. V.59. P.1279-1292.
  4. Mazandarany E.N., Peheke R.D.: ‘Thermodynamic Properties of Solid Alloys of Chromium with Nickel and Iron’ Met. Trans. 1973 4. (9) 2067-2076.
  5. Shabashov V.A., Nikolaev A.L., Mukoseev A.G., Sagaradze V.V., Filippova N.P.: ‘Mossbauer spectroscopy of thermal and radiation-rapid stratification in binary Fe-Cr alloys’ Proceedings of the Academy of Sciences, ser. phys. 2001 65 (7) 1010-1015. In Russian.
  6. Hafner J., Jaswal S.S.: ‘Interplay between atomic and electronic structure in metallic glasses: a first principles investigation’ J. Phys. F: Met. Phys. 1988 18 (1) L1-L8
  7. Andersen O.K.: ‘Linear methods in band theory’ Phys. Rev. B 1975 12 (8) 3060-3083.
  8. Kontsevoj O.Yu., Sabiryanov R.F., Mryasov O.N., Gubanov V.A.: ‘Influence of boron impurity on the electronic states and magnetic behaviour of the bcc iron: self-consistent LMTO-recursion approach’ Metallophysics 1993 15 (2) 3-8. In Russian.
  9. Almbladh C.-O., von Barth U.: ‘Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues’ Phys. Rev. B 1985 31 (9) P.3231-3244.
  10. Bratkovsky A.M., Savrasov S.Y.: ‘On the calculation of combined corrections in the LMTO method’ J. Comp. Phys. 1990 88 (1) 243-249.
  11. Aldred A.T.: ‘Ferromagnetism in Iron-Chromium Alloys. I. Bulk magnetization measurements’ Phys. Rev. B 1976 14 (1)219-227.
  12. Kajzar F., Parette G.: ‘Magnetic-moment distribution and enviromental effects in dilute iron-based alloy with V, Cr, and Mn impurities’ Phys. Rev. B 1980 22 (22), P.5471-5481.
  13. Chang Y.Y., Lin J.C., Chang Y.A.: ‘Thermodynamic description and phase relationships of the Fe-Cr system’ Calphad 1987 11 (1) 57-72.
  14. Bozorth R.M.: Ferromagnetism, New-York, 1951.
  15. Kubashevsky O.: Iron Binary Phase Diagrams, Springer, New-York, 1982.
  16. Cucera J., Million B., Ruzickova J., Foldina V., Jakobova A.: ‘Self-diffusion of Iron in α-phase of Iron and Fe-Cr alloys’ Acta Met. 1974 22 (1) 135.

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