The Role of Temperature and Magnetic Effects on the Stacking-fault Energy in Austenitic Iron
TThe Role of Temperature and Magnetic Effects on the Stacking-fault Energy inAustenitic Iron
Seyyed Arsalan Hashemi, ∗ Hojjat Gholizadeh, † and Hadi Akbarzadeh ‡ Department of Physics, Isfahan University of Technology, Isfahan, Iran (Dated: October 8, 2018)We have investigated the role of temperature and magnetic effects on the stacking-fault energy(SFE) in pure austenitic iron based on density functional theory (DFT) calculations. Using theaxial next-nearest-neighbor Ising (ANNNI) model, the SFE is expanded in terms of free energies ofbulk with face-centered cubic (fcc), hexagonal close-packed (hcp), and double-hcp (dhcp) structures.The free-energy calculations require the lattice constant and the local magnetic moments at varioustemperatures. The earlier is obtained from the available experimental data, while the later iscalculated by accounting for the thermal magnetic excitations using Monte-Carlo techniques. Ourresults demonstrate a strong dependence of the SFE on the magnetic effects in pure iron. Moreover,we found that the SFE increases with temperature.
I. INTRODUCTION
Iron and its alloys have played a significant role in thedevelopment of human civilization. Beside its industrialimportance, some unique features of iron like its phasetransitions and magnetic properties have opened inter-esting fields of research for materials scientist.The well-known phase diagram of iron shows that,at atmospheric pressure, and at low temperatures, pureiron is found in ferromagnetic body-centered cubic (bcc)structure . As temperature rises, at 1043 K (770 ℃ ), i.e. ,at the Curie temperature of iron, it demonstrates a mag-netic phase transition from the ferromagnetic to param-agnetic, while preserving the bcc structure. At 1185 K(912 ℃ ), iron faces a structural phase transition from thebcc to the face-centered cubic (fcc) structure. Furtherheating of iron reveals a second structural phase transi-tion at 1667 K (1394 ℃ ), through which the fcc structurechanges back to the bcc. Finally, at 1811 K (1538 ℃ )iron melts . Although the pure iron is found in the bccstructure at room temperatures, the addition of alloyingelements like manganese and nickel can stabilize its fccphase at room temperature .The mechanical properties of steels is influenced bytheir plastic deformations. In fcc metals, plastic deforma-tions may occur through different mechanisms includingdislocation gliding, twinning (twinning-induced plastic-ity, TWIP), and phase transformation (transformation-induced plasticity, TRIP). The activation of these mech-anisms has been proven to be governed by the stacking-fault energy (SFE) .The intrinsic stacking fault (SF) is one of the simplestplanar defects in the fcc crystal lattice . The fcc struc-ture is formed by the stacking of the close-packed layers inthe . . . ABCABCABC . . . sequence. In this structure, anISF can be considered as the elimination of a close-packedlayer in the bulk, resulting in the . . . ABC AB | AB C . . . sequence, where the removed layer has been denoted bya vertical line. This defect is demonstrated in Fig. 1.In the neighborhood of the fault, the structure resem-bles locally the stacking sequence of the hexagonal close- packed (hcp) structure, as highlighted by bold letters inthe notation (see Fig. 1). The energy associated witha SF, the SFE, is defined as the difference between thefree energy of a structure with a fault and that of theperfect fcc structure: SFE = F SF − F fcc . FIG. 1. The stacking sequence of an fcc structure along its[111] direction, with an intrinsic stacking fault at the positiondenoted by the horizontal orange line. The perfect fcc stack-ing order is highlighted in green, while the interruption dueto the fault is emphasized by the turquoise color.
There has been numerous experimental works on themeasuring of the SFE in different austenitic steels . Asalready discussed by Abbasi , Gholizadeh , and Reyes-Huamantinco , the available experimental data for theSFE are highly questionable since the reported ranges aretoo broad and have shown a high deviation from averageamount. Therefore, the development of a theoretical ap-proach based on quantum mechanics is highly desired.In recent years, the SFE and its dependence on differ-ent parameters have been the subject of many theoreticalinvestigations within the framework of the density func-tional theory (DFT) . Some of them have appliedsupercell approaches, which allow for the explicit simu-lation of the fault as well as for the relaxation of localforces, but critically restricts the calculation of magnetic a r X i v : . [ phy s i c s . c o m p - ph ] M a y or chemical disorders. For instance, Kibey et al. havestudied the dependence of the SFE on the concentrationof interstitial nitrogen in fcc iron. Similarly, Abbasi etal. and Gholizadeh et al. have studied the influenceof interstitial carbon of the SFE in fcc iron. In these threeworks, the lattice local displacements introduced by theinterstitial atom and also by the fault are accurately ac-counted for. On the other hand, using spin-unpolarizedsimulations according to 0 K equilibrium, all magneticinteractions in the paramagnetic medium are neglected.Abbasi reports only small differences between resultsobtained from tests with nonmagnetic (spin-unpolarized)and ferromagnetic calculations, supporting their simpli-fied nonmagnetic calculations where the influence of themagnetic interactions on the qualitative behavior of theSFE is assumed negligible. Referring to Abbasi’s tests,Gholizadeh states that although a nonmagnetic calcu-lation may be reliable enough to study the dependence ofthe SFE against the interstitial concentration, calculat-ing the absolute value of the SFE and also developing acomplete understanding of the atomic interactions in theparamagnetic medium requires accurately accounting forthe magnetic interactions.Other investigations have applied Green’s functionformalism , where the utilization of the disorderedlocal moments (DLM) approach and the coherent po-tential approximation (CPA) allows for the simu-lation of the magnetic and chemical disorders, respec-tively. For instance, Vitos et al. studied the tempera-ture dependence of the SFE in iron–chromium–nickel al-loys. Vitos concludes that the temperature dependenceof the SFE is almost totally dictated by the contribu-tion of the magnetic fluctuations into the free energy.Later, Reyes-Huamantinco et al. and Gholizadeh etal. improved Vitos’s approach, particularly by includ-ing the experimental data for the thermal lattice expan-sion, and calculated the temperature dependence of theSFE in iron-manganese and iron–chromium–nickel alloys,respectively. The two works reveal that the temperaturedependence of the SFE is mainly obtained from the to-tal energy of the alloy, which is in turn a function of itslattice parameter at different temperatures. Therefore,in contrast with Vitos’s results, the two works concludethat the temperature dependence of the SFE is mainlydictated by the lattice thermal expansion.Although the published works emphasize that account-ing for magnetic effects is crucial for understanding thephase stability and hence the behavior of the SFE in Fe-based alloys, an explicit comparison between quantitativeresults obtained from the paramagnetic calculations andthose obtained from nonmagnetic (spin-unpolarized) cal-culations can reveal the magnitude of the magnetic inter-actions in the SFE. Iron is the dominant element in manyindustrially interested alloys, including those mentionedabove. Further investigations show that the Fe atomsare the main responsible for the magnetic interactions inthese alloys . Moreover, performing calculations forpure iron avoids all complexities which are related to an alloy compared to an element, like atomic size mismatch,local lattice distortion, short-range and long-range or-ders, Suzuki effect, chemical phase transitions etc . There-fore, in the current work we compare two sets of theSFEs calculated for pure iron in fcc phase, one set usingthe methodology introduced by Reyes-Huamantinco etal. , and the other set using nonmagnetic calculations.Such comparison will answer two main questions: (i) Howdoes the SFE change with temperature in the paramag-netic austenitic iron? and, (ii) How big is the influenceof the magnetic effects on the SFE in the paramagneticaustenitic iron? II. METHODOLOGYA. The ANNNI Model
FIG. 2. Primitive cells of three crystal structures used in theANNNI model. Atoms are colored according to their stackingposition along the [111] direction. (a) depicts the primitivecell of the fcc structure with only one atomic site. The cu-bic cell is shown for a better imagination of the lattice. (b)shows the primitive cell of the hcp structure containing twonon-equivalent atoms (two atomic sites). (c) represents theprimitive cell of the dhcp structure with four non-equivalentatoms.
The axial next-nearest-neighbor Ising (ANNNI) model,as explained by Cheng et al. and Denteneer et al. ,expands the SFE in terms of the free energies of bulkunit-cells with different structures. In its second order,the ANNNI model results inSFE( T ) = F hcp ( T ) + 2 F dhcp ( T ) − F fcc ( T ) A + O (3) , (1)where F φ denotes the Helmholtz free energy of a singleatom in phase φ , A is the area in a close-packed layeroccupied by a single atomic site, i.e. , A = √ a , and O (3) is the error introduced by neglecting the higher or-der interactions. B. Temperature Dependence of the Free Energy
The Helmholtz free energy is defined as F ( T ) = E ( T ) − T S ( T ) , (2)where E ( T ) and S ( T ) are the total (internal) energy andentropy, respectively.The total energy E ( T ) is calculated using theDFT . Its temperature dependence originates fromthree sources: (i) the electron distribution over statesdefined by the set of occupation numbers { α (cid:15) | (cid:15) ≤ (cid:15) F } ,(ii) the lattice parameter a ( T ), and (iii) the average lo-cal magnetic moment m ( T ): E ( T ) = E (cid:0) { α (cid:15) } , a ( T ) , m ( T ) (cid:1) . (3)The Mermin functional , is applied in finite-temperatureDFT calculations to account for the temperature depen-dence of the electron distribution of over states . Thetemperature dependence of the lattice parameter is takeninto account by selecting the lattice parameters accordingto the experimental data for thermal lattice expansions(see Fig. 3). Finally, the temperature dependent local FIG. 3. The experimental lattice parameter of pure iron asa function of temperature, obtained from high-temperatureXRD measurements . magnetic moments are evaluated using a statistical ap-proach, which is explained in the next subsection. C. Longitudinal Spin Fluctuations
The thermal excitation of the local magnetic momentsin the paramagnetic DLM state is described using asimple model based on the theory of unified itinerantmagnetism . Originally, the theory accounts for bothtransverse and longitudinal spin fluctuations on equalfooting. However, here it is further simplified so thatthe transverse spin fluctuations, i.e. , the fluctuations inthe orientations of the magnetic moments, always followthe completely disordered configuration described in theDLM state. The longitudinal spin fluctuations (LSF), i.e. , the fluctuations in the size of the magnetic moments,are obtained by performing a classical Monte-Carlo sim-ulation over a mapping of the system energetics. Thesesystem energetics are calculated using the EMTO code.For representing the energy of the classical magneticstate, Ruban et al. introduced a magnetic Hamiltonianwhich was later applied to an Fe–Mn alloy by Reyes-Huamantinco et al. . In the case of pure iron, the Hamil-tonian is simplified as H mag . ( m ) = J ( m ) , (4)where m is the spatially-averaged local magnetic mo-ments of iron atoms, and J ( m ) is the energy required toexcite this averaged moment from 0 to the value m in theDLM paramagnetic state. After calculating the Hamilto-nian parameters, J ( m ), a Metropolis Monte-Carlo tech-nique is applied to find the temperature-dependence ofthe local magnetic moments . D. Entropy Contributions
In a solid, the entropy consists of configurational, vi-brational, magnetic, and electronic contributions : S = S vib . + S conf . + S mag . + S el . , (5)where all terms are associated to a single site in the lat-tice.Currently, there are no available theoretical tools todetermine the vibrational entropy in paramagnetic ran-dom alloys.The configurational entropy is connected to the disor-der in the arrangement of atoms of different species in thematerial. Therefore, in a pure element where all atomsare equal, it simply vanishes: S conf . = 0.For the ideal paramagnetic state, the magnetic entropyis evaluated using the mean-field expression S mag . ( T ) = k B ln (cid:0) m ( T ) + 1 (cid:1) . (6)Here, m ( T ) denotes the average (over time and space)local magnetic moment obtained using the mentionedMonte Carlo calculations.The electronic entropy can be calculated as S el . ( T ) = − k B (cid:90) (cid:110) f ( (cid:15) ) ln (cid:0) f ( (cid:15) ) (cid:1) (7)+ (cid:0) − f ( (cid:15) ) (cid:1) ln (cid:0) − f ( (cid:15) ) (cid:1)(cid:111) D ( (cid:15) ) d (cid:15), where, D ( (cid:15) ) and f ( (cid:15) ) denote the density of states and thefinite-temperature Fermi function , respectively. TheFermi function is a consequence of the Fermi-Diracstatistics and gives the probability of occupation ofa state with energy (cid:15) at temperature T . The Fermi func-tion is defined as f ( (cid:15) ) = 1 e ( (cid:15) − µ ) /k B T + 1 , (8)where µ indicates the chemical potential. E. The SFE Calculations
In order to find the total energy as a function of tem-perature, the temperature-dependent lattice parameterand local magnetic moments are used in a set of con-strained
DFT calculations, where the magnetic momentis fixed to its corresponding value evaluated using theMonte-Carlo method. The free energy is simply foundby subtracting the entropy contributions from the totalenergy (see Eqs. 2 and 5). The SFE is evaluated usingthese total energies, according to the Eq. 1.
III. CONCLUSION
Our results evaluated using both spin-unpolarized andspin-polarized calculations show that the SFE increaseswith temperature. Although, spin-polarized calculationssuggest that the temperature dependence of the SFE van-ishes at high temperatures. The similar trend of the SFEin both calculations shows that the thermal dependenceof the SFE is mainly originated from the thermal lat-tice expansion. This is in contrast with the publishedwork by Vitos et al. , but agrees with results found byReyes-Huamantinco and Gholizadeh . We also findthat, in order to get the correct phase stability of ironand also the correct value of its SFE, one must accountfor magnetic interactions in the system, including thespin fluctuations. ACKNOWLEDGMENTS ... ∗ [email protected] † [email protected] ‡ [email protected] T. B. Massalski, H. Baker, L. H. Bennett, and J. L. Mur-ray,
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