Effect of nitrogen on stacking fault energy of f.c.c. iron-based alloys

Effect of nitrogen on stacking fault energy of f.c.c. iron-based alloys

PII: Acta mater. Vol. 47, No. 4, pp. 1271±1279, 1999 # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in ...

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PII:

Acta mater. Vol. 47, No. 4, pp. 1271±1279, 1999 # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00419-4 1359-6454/99 $19.00 + 0.00

EFFECT OF NITROGEN ON STACKING FAULT ENERGY OF F.C.C. IRON-BASED ALLOYS I. A. YAKUBTSOV, A. ARIAPOUR and D. D. PEROVIC{ Department of Metallurgy and Materials Science, University of Toronto, Toronto, Ontario M5S 3E4, Canada (Received 1 October 1998; accepted 24 November 1998) AbstractÐA method is proposed to estimate the stacking fault energies of face-centered-cubic (f.c.c.) ironbased alloys. The segregation of alloying elements to stacking faults and the interaction of substitutional and interstitial alloying elements in solid solution and their e€ect on stacking fault energy have been taken into account. It is shown that at low nitrogen concentrations (e.g. 0.05 wt%), the stacking fault energy is increased mainly due to the e€ect arising from the bulk of the alloy. At high nitrogen concentration (e.g. 0.5 wt%), the stacking fault energy is decreased due to the segregation of the alloying elements (mainly nitrogen) on the stacking faults of the alloy. Moreover, it is shown that in nitrogen alloying of f.c.c. ironbased alloys the magnetic contribution of the nitrogen to the stacking fault energy is negligible. The method shows reasonable agreement with existing experimental data. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

The stacking fault energy of a material is an important characteristic since it dictates many physical properties of the material such as phase stability and transformation [1±3], mechanical behavior [4] and stress corrosion cracking [5]. Due to the technological importance of ironbased alloys, the stacking fault energy of this system has attracted the attention of many researchers [6±9]. Despite thorough investigations on the e€ect of temperature and alloying elements on the stacking fault energy of face-centered-cubic (f.c.c.) iron-based alloys [7±9] there are some discrepancies in the literature regarding the e€ect of nitrogen on the stacking fault energy of f.c.c. ironbased alloys. It has been shown experimentally [10, 11] that nitrogen increases the stacking fault energy of stainless steels with f.c.c. structure. However, other experiments [12±14] have shown that nitrogen decreases the stacking fault energy of f.c.c. iron-based alloys. Many attempts have been made to ®nd a theoretical method for estimating the stacking fault energy of f.c.c. alloys. Generally two methods have been employed, ``The Electronic Models [15]'' and ``The Thermodynamic Treatments [16±19]''. Petrov and Yakubtsov [19] developed a method based on statistical thermodynamics to calculate the stacking fault energy of multi-component alloys with f.c.c. structures having both substitutional and interstitial alloying elements. Their calculation considered the interaction between substitutional and interstitial {To whom all correspondence should be addressed.

alloying elements in solid solution and the segregation of the alloying elements to the faults. Their method of calculation of stacking fault energy showed good agreement with experimental results for an iron alloy having carbon as the interstitial alloying element [19]. In their work it was shown theoretically and experimentally that carbon decreased the stacking fault energy of the f.c.c. iron alloys at low concentrations but increased the stacking fault energy of the alloy system at high carbon concentrations (see Fig. 1). In this work we present a similar thermodynamic approach to Petrov and Yakubtsov [19] for calculating the stacking fault energy of a quaternary f.c.c. iron-based alloy (Fe±18Cr±10Ni±XN) having nitrogen as the principal interstitial alloying element. We ®rst present the method of our calculations, which has been constructed based on a regular solid solution model. We then expand our method to calculate the stacking fault energy of an Fe±Cr±Ni±Mn±N alloy and compare our calculations with experimental results published in the literature [20]. It will be shown that there is good compatibility between the calculated results in this work and the published experimental results. At the end of the discussion section the limitation of our model, which is based on a regular solid solution approximation, will be discussed. 2. STACKING FAULT ENERGY IN F.C.C. STRUCTURES

Stacking faults in the f.c.c. structure are in fact thin plates of h.c.p. structure (i.e. three or two atomic monolayers for intrinsic or extrinsic stacking

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stacking fault energy. The above de®nition can be rewritten in terms of free energy change as follows: Zˆ

1 4e 4e 4e …DG gÿ ‡ DG gÿ ‡ DG gÿ s m †: b 8:4V 2=3

…4†

Accordingly, to estimate the stacking fault energy we need to calculate the di€erent free energy contributions for a g 4 e phase transformation in our alloy system. 3. BULK FREE ENERGY CHANGE OF F.C.C. (gg) 4 H.C.P. (ee) PHASE TRANSFORMATION

Fig. 1. Stacking fault energy of a f.c.c. Fe±22Mn±XC alloy vs carbon concentration. Curve (a) is based on experimental results of stacking fault energy measurements, and curve (b) is based on calculated values of stacking fault energy [18].

faults, respectively). Hence, the stacking fault energy in a f.c.c. structure could be written based on the free energy change due to the f.c.c. (g) 4 h.c.p. (e) phase transformation such that: Zˆ

DG gÿ4e V=t

…1†

where Z is the stacking fault energy, DGg 4 e is the free energy di€erence between the e and g phases, V is the molar volume of alloy, and t is the thickness of the fault. By assuming that the e€ect of thickness on stacking fault energy is negligible (since e forms as very thin plates in g), equation (1) in ``SI'' units becomes [17]: 1 Zˆ DG gÿ4e : 8:4V 2=3

…2†

Ericsson [21] separated the stacking fault energy of a system into three parts: Z ˆ Zb ‡ Zs ‡ Zm

…3†

where Zb is the stacking energy di€erence between the f.c.c. and h.c.p. structures per unit area for bulk material, Zs is the energy which arises from the concentration di€erence of alloying elements in the fault region and in the matrix (i.e. the segregation e€ect), and Zm is the magnetic contribution to the {In a binary alloy the interaction parameter (W) for g and e phases is de®ned as [26]: GE ˆ X…1 ÿ X †W, where GE is the excess free energy due to the g 4 e transformation for each phase. The free energy change for the transformation is expressed as: DGg4e 4e gÿ 4e ˆ …1 ÿ X †DG gÿ ‡ G eE ÿ G gE . …1 ÿ X † ‡ xDG X

Our ®rst assumption for calculating the bulk free 4e energy change, DG gÿ b , is that the system is a regular solid solution of: (i) substitutional (sub) atoms; and (ii) interstitial (int) atoms in octahedral sites in the f.c.c. and h.c.p. structures. The atomic con®gurations in the structure, such as vacancies [22, 23], are not considered in this model due to unavailability of experimental data for our system, which are required for calculation. Therefore, we can write 4e gÿ 4e DG bgÿ4e ˆ DG gÿ b…sub† ‡ DG b…int†

…5†

4e where DG gÿ b…sub† is the bulk free energy change due to substitutional atoms for a g 4 e transformation, 4e and DG gÿ b…int† is the bulk free energy change due to interstitial atoms for a g 4 e phase transformation. 4e 3.1. Calculation of DG gÿ b…sub†

To calculate the free energy change arising from the substitutional atoms we employ a regular solid 4e solution model. Therefore, the DG gÿ b…sub† contribution can be expressed in classical thermodynamic form as follows [17]: 4e DG gÿ b…sub† ˆ

n X 4e X…sub†i DG gÿ …sub†i i

‡

n X n 1X 4e DW gÿ Xi Xj ij 2 i j

…6†

where X…sub†i is the atomic percentage of the substi4e tutional component of type ``i'', DG gÿ …sub†i is the free energy change due to the g 4 e transformation arising from the substitutional component of type ``i'', 4e and DW gÿ (i 6ˆ j) is the di€erence of interaction ij parameters{ between ``i'' and ``j'' substitutional components due to the g 4 e phase transformation. 4e 3.2. Calculation of DG gÿ b…int†

Many studies have been performed to evaluate the thermodynamic behavior of interstitial elements in alloy systems using classical and statistical thermodynamic approaches [24±26]. The statistical thermodynamic approaches have been more successful [25] and as such have been employed here. We de®ne the bulk free energy change arising from interstitial atoms for g and e phases in terms of con®gurational energy of the substitutional

YAKUBTSOV et al.: STACKING FAULT ENERGY

atoms surrounding the interstitial alloying elements. 4e Consequently, DG gÿ b…int† can be expressed as 4e g e DG gÿ b…int† ˆ E N ÿ E N

…7†

where E eN and E gN are the con®gurational energies (or internal energies{) for interstitial atoms (e.g. nitrogen) in the e and g phases, respectively. In order to calculate E eN and E gN values as in previous sections we again employ the approximation for a regular solid solution. In statistical thermodynamics this approximation implies that: (a) the interaction energy between the interstitial (e.g. nitrogen) and substitutional atoms only arises from the ®rst nearest neighbors; (b) the lattice parameter of the structure is independent of temperature and composition, therefore, the interaction energy between any two substitutional atoms or substitutional and interstitial atoms, which is a function of lattice parameter, will be independent of temperature and composition; (c) all the alloying elements are distributed homogeneously; (d) the interaction energy between the interstitial atoms is equal to zero in solid solution. The bulk con®gurational energy of the quaternary f.c.c. alloy system considered in this work, which has three components occupying the substitutional sites, and one interstitial alloying element (N) occupying only the octahedral sites (i.e. six immediate neighbors for each site), can be expressed as follows: E gN ˆ

6 …6ÿkÿj 6 X †…6ÿk† XX X ‰…n1 †i U g1N ‡ …n2 †j U g2N

mˆ0

iˆ0

‡ …n3 †k U

jˆ0 kˆ0

g 3N Š…nN †m

…n1 †i ‡ …n2 †j ‡ …n3 †k ˆ 6 6 X mˆ0

…nN †m ˆ NN

bulk of the alloy, ‰U g1N ,U g2N ,U g3N Š are interaction energies between each type of substitutional atom and interstitial element (N) in the g phase, m is the number of octahedral sites in each unit cell of the f.c.c. structure, nN is the number of nitrogen atoms in interstitial (i.e. octahedral) sites in each f.c.c. unit cell, and NN is the total number of nitrogen atoms in the bulk material. By substituting (n2)j with …6 ÿ …n1 †i ÿ …n3 †k † in equation (8) we obtain: E gN ˆ

6 …6ÿkÿj 6 X †…6ÿk† XX X ‰…n1 †i U g1N ‡ f6 ÿ …n1 †i

mˆ0

iˆ0

jˆ0 kˆ0

ÿ …n3 †k gU g2N ‡ …n3 †k U g3N Š…nN †m

E gN ˆ …U g1N ÿ U g2N †

‡ 6U g2N

6 X

4e ˆ E e ÿ E g ‡ …ST †e ÿ …ST †g : DG gÿ

It should be mentioned that both the number of substitutional sites around a nitrogen atom and the number of octahedral sites in each cell of a f.c.c. are the same as the h.c.p. structure. Therefore, the entropy terms in both phases are equal (that is) …ST †e ˆ …ST †g . Subsequently we 4e get: DG gÿ ˆ E e ÿ E g.

iˆ0

…n1 †i …nN †m

6 X 6 X …n3 †k …nN †m

mˆ0 kˆ0

…nN †m :

…12†

Smirnov [27] showed that, in an alloy system with the close packed structure (i.e. f.c.c. and h.c.p.), if there are many types of substitutional alloying elements …1, . . . ,z† and only one type of interstitial alloying element which occupies the octahedral sites the following equation are valid:

…9†

{The free energy is de®ned as G ˆ E ‡ ST and S ˆ K ln… p†, where p is the number of con®gurations of substitutional and interstitial atoms. For a g 4 e phase transformation we can write:

mˆ0

…11†

mˆ0

U g1N RT …n1 †i …nN †m ˆ X U gZN mˆ0 iˆ0 XZ exp RT Z

…13†

X XZ ˆ 1

…14†

6XN X1 exp

6 X X

where ‰n1 ,n2 ,n3 Š are the types of substitutional elements, ‰…n1 †i ,…n2 †j ,…n3 †k Š are numbers of atoms of each type occupying the substitutional sites in the

6 …6ÿkÿj X† X

‡ …U g3N ÿ U g2N †

…8†

…10†

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and

Z

where X1 and XZ are atomic percent of substitutional atoms of type ``1'' and ``Z'' (respectively), Z is the type of substitutional alloying element, XN is the atomic percent of nitrogen in the alloy, and U g1N and U gZN are the interaction energies of nitrogen atoms with substitutional atoms of type ``1'' and ``Z'' (respectively). Therefore, in an alloy system with three types of elements in substitutional sites and one type of atom in interstitial sites of octahedral type,

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equation (12) can be written as: E gN ˆ 6XN U g2N ‡ …U g1N U g1N RT ‡ …U g ÿ U g2N † 3 3N X U gZN XZ exp RT Zˆ1 6XN X1 exp

U g3N RT : ÿ U g2N † 3 X Ug XZ exp ZN RT Zˆ1 6XN X3

…15†

The e€ect of nitrogen on the free energy change for the f.c.c. 4 h.c.p. phase transformation arises from the di€erence between the con®gurational energy of nitrogen in f.c.c. and h.c.p. lattices. The con®gurational energy of nitrogen in the h.c.p. structure can be calculated in a similar way as presented in equations (8)±(15) for the f.c.c. structure (see footnote on p. 1273). However, the only di€erence will be in the interaction energy of the interstitial atoms and the substitutional atoms. Finally, the di€erence between the con®gurational energies of two systems (g and e) arising from the presence of nitrogen can be expressed as:

In calculating the stacking fault energy of a f.c.c. alloy Ishida [17] assumed that the free energy change arising from segregation of the alloying elements to stacking faults is negligible. His reasoning was based on the fact that the DGchm and DGels are of similar magnitude but have opposite signs and will cancel each other while the DGsur contribution is negligible. Therefore, Ishida constructed his calculations for stacking fault energy estimation based on ``bulk'' and ``magnetic'' contributions to the stacking fault energy, and he ignored the segregation e€ect of alloying elements on stacking fault energy. However, the large amount of existing data that describe the strong interaction of interstitial alloying elements with dislocations and the associated segregation of alloying elements to stacking faults [28±31] prevent us from ignoring the e€ect of these phenomena on stacking fault energy. 4.1. Calculation of DGchm For calculating the DGchm values we consider the e€ect of the interstitial and substitutional alloying elements separately such that DGchm ˆ DGchm…sub† ‡ DGchm…int† :

…18†

In order to calculate each term of equation (18) we

4e g e DG gÿ b…int† ˆ E N ÿ E N

 ˆ 6XN …U e2N ÿ U g2N † ‡

‡

ÿ

ÿ

 X1 ‡ X2 exp ÿ

…U

…U e1N ÿ U e2N †X1    ÿ U e2N † …U e1N ÿ U e3N † ‡ X3 exp ÿ RT RT

e 1N

…U e3N ÿ U e2N †X3     e e …U 3N ÿ U 1N † …U e3N ÿ U e2N † ‡ X2 exp ÿ ‡ X3 X1 exp ÿ RT RT …U g1N ÿ U g2N †X1     g …U g1N ÿ U g3N † …U 1N ÿ U g2N † ‡ X3 exp ÿ X1 ‡ X2 exp ÿ RT RT  …U g3N ÿ U g2N †X3     : …U g3N ÿ U g1N † …U g3N ÿ U g2N † ‡ X2 exp ÿ ‡ X3 X1 exp ÿ RT RT

4. FREE ENERGY CHANGE DUE TO SEGREGATION OF ALLOYING ELEMENTS ON STACKING FAULTS

As shown by Ishida [17], the free energy changes 4e due to segregation (DG gÿ ) of alloying elements on s stacking faults may be divided into three parts: 4e DG gÿ ˆ DGchm ‡ DGsur ‡ DGels s

…17†

where DGchm is the chemical free energy due to Suzuki segregation, DGsur is the surface free energy due to the concentration di€erence between the matrix and the stacking faults, and DGels is the elastic free energy which is due to the segregation of elements having di€erent atomic sizes.

…16†

need to know the concentration of the alloying elements on faults. When the equilibrium condition is satis®ed, we can assume that the chemical potential of the g phase is the same as the chemical potential of the e phase such that dG g @ G e ˆ dXb @ Xs

…19†

where Gg and Ge are the free energy of g and e phases, and Xb and Xs are concentrations of the alloying elements in the bulk and on the stacking faults, respectively. Ishida [17] calculated the concentration of alloying elements on stacking faults of binary alloy sys-

YAKUBTSOV et al.: STACKING FAULT ENERGY

tems using equation (19) by assuming that the systems were ideal solid solutions (i.e. U gij ˆ U eij ˆ 0). Ishida [17] had to make this assumption due to the fact that the interaction energies of the alloying elements of interest for calculation (e.g. U gFe,Mn ) were not available in the literature. For the same reason in this part of the work we also employed Ishida's assumption and then derived (see the Appendix A) an equation to estimate the Xs values for a multicomponent system as follows: " X es…i † ˆ 1 ‡

#  4e 4e  ÿ1 DG gÿ ÿ DG gÿ j i exp : RT X gb…i †

X X gb…i † j

…20† Since all the thermodynamic data required in equation (20) are not available for nitrogen, we simplify equation (20) to give Xs…N †

   …1 ÿ Xb…N † † ÿL ÿ1 ˆ 1‡ exp Xb…N † RT

…21†

where L is the interaction energy of the interstitial element (i.e. nitrogen) with dislocations in the f.c.c. structure. We calculate the DGchm according to Ref. [17], for both substitutional and interstitial alloying elements in our studies using DGchm ˆ RT

X i

X…b†i ln

Xs…i † : Xb…i †

…22†

4.3. Calculation of DGels Suzuki [30] showed that the elastic energy, which is attributed to the segregation of elements with di€erent atomic sizes, could be described by   2 1 ‡  dV 1 DGels ˆ m …Xs ÿ Xb †2 9 1 ÿ  dX V

The surface free energy change of stacking faults due to the segregation of alloying elements, deduced from work of Erricsson [21], is expressed by equation (23){ …23†

where L is the interaction energy of the alloying element with the stacking fault, and Xs and Xb are the concentrations of the alloying element on the stacking fault and in the bulk of the material, respectively. In the case of substitutional alloying elements equation (23) predicts negligible values, therefore we have only taken into account the surface segregation of interstitial alloying elements. {DGsur should always take a negative sign for equilibrium segregation to occur. Gsur for a new state of segregation should be less than the Gsur for the initial state of segregation. {The magnetic free energy arising from substitutional alloying elements for h.c.p. structures is taken to be equal to zero in this work [17].

…24†

where m is shear modulus, n is Poisson's ratio, and V is the molar volume of the alloy. The elastic free energy changes for both the substitutional and the interstitial alloying elements in the system of interest in this work were estimated to be negligible [17] and are therefore ignored here.

5. EFFECT OF NITROGEN ON THE MAGNETIC CONTRIBUTION TO FREE ENERGY CHANGE

The total magnetic free energy change for substitutional alloying elements is calculated from the following expression [17]:{ 4e DG gÿ ˆ m

i i X X 4e DG gÿ X ˆ ÿ G g…m†i Xi : i …m†i iˆ1

…25†

iˆ1

Tauer and Weiss [32] formulated the magnetic free energy of a system as a function of the magnetic transformation temperature (Tc) and the number of Bohr magnetons (2S) as follows:  4 T G gm ˆ ÿRTc S 1=3 ÿ RTc ‰0:4 ln…2S ‡ 1† Tc ÿ 0:2S 1=3 Š for TRTc

4.2. Calculation of DGsur

1 DGsur ˆ L…Xs ÿ Xb †2 4

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…26†

G gm ˆ ÿRT ln…2S ‡ 1† ÿ RT 2c ‰0:8S 1=3 ÿ 0:6 ln…2S ‡ 1†ŠT ÿ1 for TrTc :

…27†

The values for Tc and 2S for many metals such as Fe, Mn, Cr, and Ni are available in the literature [18]. However, these values are not available when nitrogen is included as an alloying element. Frisk [23] used the Hertzman and Sundman [33] expression to calculate the magnetic free energy of the Fe±Ni±N system. This expression is Gm ˆ RT ln…b ‡ 1† f…t†

…28†

where b is the average magnetic moment per atom (in Bohr magnetons), and t is T/Tc. In equation (28), f(t) is de®ned as [30±32]   3  79tÿ1 474 1 t t9 t15 ÿ1 ‡ ‡ ‡ 140p 497 p 6 135 600 f …t† ˆ 1 ÿ A for t<1

…29†

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f …t† ˆ ÿ where

YAKUBTSOV et al.: STACKING FAULT ENERGY ÿ5

ÿ15

t t tÿ25 ‡ ‡ 10 315 1500 A



Table 1. Thermodynamic values used in this work

for t > 1

…30†

4e DGgÿ Fe 4e DGgÿ Cr 4e DGgÿ Ni gÿ 4e



 518 11692 1 Aˆ ‡ ÿ1 1125 15975 p

and the parameter p which depends on crystal structure is equal to 0.4 for b.c.c. and 0.28 for f.c.c. or h.c.p. structures. Based on the work of Frisk [23] and our estimation, the magnetic free energy changes of an iron alloy with 30 wt% Ni and concentration of 0± 0.5 wt% N for both f.c.c. and h.c.p. structures are small (i.e. 01 mJ/m2). Accordingly, nitrogen has a minor e€ect on increasing the magnetic free energy. Therefore, in our calculations we only consider the e€ect of substitutional alloying elements on magnetic free energy changes using equations (25)±(27). 6. ESTIMATION OF STACKING FAULT ENERGY OF Fe±18Cr±10Ni SYSTEM ALLOYED WITH NITROGEN

Following the methodology presented in the previous sections we have estimated the stacking fault energy of the Fe±18Cr±10Ni system alloyed with varying levels of nitrogen. Based on the assumptions we have made, to cal4e culate the stacking fault energy we require DG gÿ b , gÿ 4e gÿ 4e DG s , and DG m values. To calculate the ®rst two terms we require: 1. the di€erence in interaction energies between each pair of nitrogen and substitutional alloying elements in f.c.c. and h.c.p. structures; and 2. interaction energies of nitrogen with a stacking fault. According to Kaufman and Bernstein [34] 4e U eij ˆ DG gÿ ‡ U gij i

…31†

where ``i'' is the major alloying element. Therefore, for our system we can write: …U eFe±N ÿ U eNi±N † 4e gÿ 4e ˆ …U gFe±N ÿ U gNi±N † ‡ …DG gÿ Fe ÿ DG Ni †

…32†

…U eFe±N ÿ U eCr±N † 4e gÿ 4e ˆ …U gFe±N ÿ U gCr±N † ‡ …DG gÿ Fe ÿ DG Ni †

Thermodynamic term

DGMn gÿ 4e DWFe gÿ 4±e Ni DWFe ± Cr gÿ 4e DWNi gÿ 4±eCr DWFe ± Mn DWgNi ± Mn UgFe ± N ÿ UgCr ± N UgFe ± N ÿ UgNi ± N UgCr ± N ÿ UgNi ± N DGg…m†Fe DGg…m†Cr DGg…m†Ni

Numerical value (J/mol)

Ref.

ÿ314 ÿ2284 1425 3478 2095 2095 4190 ÿ6704 0 18 800 ÿ17 000 ÿ35 800 ÿ888 0 ÿ3553

[34] [34] [34] [24] [26] [26] [26] [24] [24] [25] [25] [25] [17] [17] [17]

system. In order to calculate the e€ect of nitrogen segregation to stacking faults we require the interaction energy values of nitrogen atoms with stacking faults. Since the exact values of interaction energies of nitrogen with stacking faults or partial dislocations are not available, we have taken the interaction energy values of nitrogen atoms with dislocations in the f.c.c. structure from the work of Gavrilyuk et al. [29] as the closest approximation in order to solve the problem. Gavrilyuk et al. [29] performed internal friction tests to determine the interaction energies of nitrogen with dislocations in a f.c.c. steel structure and showed a nitrogen concentration dependency in the range 0.13±0.52 wt% nitrogen. We have used the result of their experimental data and extrapolated those values to the range of 0±0.13 wt% of nitrogen. The ®rst three values in Table 2 were obtained by extrapolating the data presented in Ref. [29]. The results of our calculations are shown in Figs 2±4. Figure 2 presents the concentration of N on stacking faults (X(s)N) vs the concentration of nitrogen in the bulk of the alloy (X(b)N). It can be seen that by increasing the concentration of nitrogen in the bulk of the material the concentration on stacking faults is also increased. In Fig. 3 we present the two contributions to the stacking fault energy which arise from the bulk of the alloy due to nitrogen addition, Z(b)N, and segregation of nitrogen to the stacking faults, Z(s)N, as a

…33†

…U eNi±N ÿ U eCr±N † 4e gÿ 4e ˆ …U gNi±N ÿ U gCr±N † ‡ …DG gÿ Ni ÿ DG Cr †: …34†

Table 1 summarizes values for free energy changes and the interaction energies and parameters obtained from the literature. Table 2 contains the interaction energy of nitrogen atoms with dislocations in a f.c.c. iron-based

Table 2. E€ect of nitrogen concentration on interaction energy of nitrogen with dislocations in a f.c.c. iron alloy N (wt%) 0.00 0.05 0.10 0.13 0.25 0.36 0.52

LN (J/mol)

Ref.

0.0 1537.2 2964.6 3568.5 3788.1 4831.2 6862.5

Ð Ð Ð [29] [29] [29] [29]

YAKUBTSOV et al.: STACKING FAULT ENERGY

Fig. 2. Calculated segregation concentration of nitrogen to stacking faults (Xs(N)) vs concentration of nitrogen in the bulk of the material (Xb(N)), in atomic percentage.

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Fig. 4. Calculated stacking fault energy vs nitrogen concentration for (a) Fe±18Cr±10Ni±XN alloy system, and (b) Fe±18Cr±10Ni±8Mn±XN.

function of nitrogen concentration in the alloy system. Figure 3 shows that nitrogen increases the Z(b)N values while it decreases the Z(s)N values of the system. In curve (a) of Fig. 4 we present the total stacking fault energy of the Fe±18Cr±10Ni±XN system vs nitrogen concentration. It can be seen that the calculated stacking fault energies for Fe±18Cr± 10Ni±XN alloy are of the same order as published experimental data [7]. Moreover, the stacking fault

energy is observed to increase to a maximum value and then to decrease as nitrogen increases. In curve (b) of Fig. 4 we present the results of our calculations for a f.c.c. iron-based system with four substitutional alloying elements plus nitrogen (i.e. Fe±18Cr±10Ni±8Mn±XN) in the same way as for the Fe±18Cr±10Ni±XN alloy system. Figure 5 shows the experimental results of Fujikura et al. [20] on the e€ect of nitrogen on stacking fault energy of iron-based alloys. These authors showed that the

Fig. 3. Calculated bulk (Z(b)N) and segregation (Z(s)N) contributions to the stacking fault energy arising from nitrogen addition to the alloy system vs nitrogen concentration.

Fig. 5. Experimental results for the probability of stacking fault vs nitrogen concentration for a Fe±18Cr±10Ni± 8Mn±XN alloy system [20].

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YAKUBTSOV et al.: STACKING FAULT ENERGY

probability of stacking fault formation for Fe± 18Cr±10Ni±8Mn±XN alloy decreases for nitrogen levels <0.14 wt% and then it is followed by an increase. It is seen that our method is capable of predicting the same trend for the e€ect of nitrogen on the stacking fault energy of a complex alloy as the experimental results. Upon comparison of curves (a) and (b) of Fig. 4 it is seen that the addition of manganese (8 wt%) to the Fe±18Cr±10Ni±XN alloy has increased the stacking fault energy of the alloy system. In comparing the experimental results with the results based on our calculations one should keep in mind that the real stacking fault energy of an alloy is a€ected by many factors, which cannot be easily formulated and were not included in this work. One of the major factors is the thermal history of the alloy, which a€ects the microstructure (e.g. grain size, twins, and interface boundaries) of the alloy, and therefore the concentration of the alloying elements in solid solution. By applying di€erent types of heat treatment to an alloy the concentration of the alloying elements, especially interstitial alloying elements, will be di€erent due to the segregation of alloying elements on the boundaries and interfaces. The method presented here can be improved in order to estimate closer values to the real stacking fault energy of the multi-component f.c.c. ironbased system alloyed with nitrogen if: 1. the interaction parameter between interstitial atoms in both f.c.c. and h.c.p. phases is included in the calculations; 2. the interaction energy of nitrogen atoms with dislocations in f.c.c. structure is used for low concentration of interstitial atoms (the extrapolated values for N concentrations <0.13 wt% were used here); 3. the clustering e€ect of substitutional alloying elements and the ordering e€ect of nitrogen in the bulk of the material is considered in the calculation. 7. CONCLUSIONS

In this work we have developed a method to estimate the stacking fault energy from the free energy changes of the system. 1. The method is capable of predicting the e€ect of nitrogen on the stacking fault energy of ironbased alloys with the f.c.c. structure. Two parts dictate the value of stacking fault energy at each nitrogen concentration: the stacking fault energy arising from the bulk, Zb; and the stacking fault energy arising from segregation of alloying elements to stacking faults, Zs. At low levels of nitrogen the Zb contribution plays the major role to increase Z; however, at high levels of nitrogen Z decreases due to the major e€ect of Zs. This

e€ect is di€erent from the e€ect of carbon (cf. Fig. 1 with Fig. 4). 2. The nitrogen concentration for maximum stacking fault energy in f.c.c. structure depends on the interaction energy of nitrogen with stacking faults, the interaction energy of nitrogen with substitutional alloying elements in solid solution, and the concentration of substitutional alloying elements in solid solution. 3. The in¯uence of nitrogen on the magnetic part of the stacking fault energy is small.

AcknowledgementsÐThe authors appreciate the ®nancial support of the Natural Sciences and Engineering Research Council (NSERC) and the Ontario Centre for Materials Research (OCMR) of Canada through completion of this work.

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YAKUBTSOV et al.: STACKING FAULT ENERGY 27. Smirnov, A. A., Molekulyarno-kineticheskaya teoriya metallov (The Molecular Kinetic Theory of Metals). Nauka, Moscow, 1966. 28. Herschitz, R. and Seidman, D. N., Acta metall., 1985, 33(8), 1547. 29. Gavrilyuk, V. G., Duz, V. A., Ye®menko, S. P. and Kvasnevskiy, O. G., Physics Metals Metallogr., 1987, 64(6), 84. 30. Suzuki, H., Dislocation and Mechanical Properties of Crystals. Wiley, New York, 1957. 31. Kamino, T., Ueki, Y., Hamajim, H., Sasaki, K., Kuroda, K. and Saka, H., Phil. Mag. Lett., 1992, 66(1), 27. 32. Tauer, K. J. and Weiss, R. J., J. Phys. Chem. Solids, 1958, 4, 135. 33. Hertzman, S. and Sundman, B., CALPHAD, 1982, 6, 67. 34. Kaufman, L. and Bernstein, H., Computer Calculation of Phase Diagram. Academic Press, New York, 1970.

dG g d ˆ ÿG gA ‡ G gB ‡ RT dX…b†B dX g…b†B  ‰…1 ÿ X g…b†B ÿ X g…b†C †  ln…1 ÿ X g…b†B ÿ X g…b†C † ‡ X g…b†B ln X g…b†B ‡ X g…b†C ln X g…b†C Š

To determine the concentration of the alloying elements on stacking faults one should assume that stacking faults in f.c.c. structures are thin plates of h.c.p. phase. Therefore, in equilibrium condition: dG gB @ G eB ˆ : dX g…b†B @ X e…s†B

‡ RT‰ÿln…1 ÿ X g…b†B ÿ X g…b†C † ÿ 1 ‡ ln X g…b†B ‡ 1Š

…A7†

X g…b†B dG g ˆ ÿG gA ‡ G gB ‡ RT ln : g dX …b†B 1 ÿ X g…b†B ÿ X g…b†C

X e…s†B @G e ˆ ÿG eA ‡ G eB ‡ RT : e @ X …s†B 1 ÿ X e…s†B ÿ X e…s†C

‡ RTX

ln X

g …b†A

‡ RTX

g …b†B

ln X

ÿ G gA ‡ G gB ‡ RT ln ÿ G eA ‡ G eB ‡ RT ln

X g…b†B

1 ÿ X g…b†B ÿ X g…b†C

ˆ

X e…s†B

1 ÿ X e…s†B ÿ X e…s†C

or 4e 4e ÿDG gÿ ‡ DG gÿ ‡ RT ln B A

X e…s†C …1 ÿ X g…b†B ÿ X g…b†C † …1 ÿ X e…s†B ÿ X e…s†C †X g…b†B

…A2†

X g…b†A ‡ X g…b†B ‡ X g…b†C ˆ 1

…A3†

X g…b†A ˆ 1 ÿ X g…b†B ÿ X g…b†C

…A4†

4e 4e ÿDG gÿ ‡ DG gÿ ‡ RT ln B A

where

W

ˆW

g AC

X e…s†C …1 ÿ X g…b†B ÿ X g…b†C † …1 ÿ X e…s†B ÿ X g…s†C †X g…b†C

…A12†

1   4e gÿ 4e  4e 4e  X g…b†C DG gÿ ÿ DG DG gÿ ÿ DG gÿ B B C A 1 ‡ g exp ‡ g exp RT RT X …b†B X …b†B

ˆW

ˆ 0:

ˆ 0:

By solving equations (A11) and (A12) as a set of equations with two unknown variables (i.e. X e…s†B and X e…s†C ) we get the equations, such as equation (A13), to calculate the values of unknown variables

X g…b†A

g BC

ˆ 0: …A11†

‡ X g…b†A X g…b†C W gAC ‡ X g…b†B X g…b†C W gBC

g AB

…A10†

Now, if we get the ®rst derivative of equation (A2) with respect to XC in the same way as equations (A5), (A6), (A7), (A8), (A9) and (A10) we get the following equation:

g …b†B

‡ RTX g…b†C ln X g…b†C ‡ X g…b†A X g…b† W gAB

X e…s†B ˆ

…A9†

Since the left-hand terms of equations (A8) and (A9) are equal we will have

G g ˆG gA X g…b†A ‡ G gB X g…b†A ‡ G gC X g…b†B g …b†A

…A8†

The left-hand term of equation (A8) is the same as the left-hand term of equation (A1). To derive the right-hand term in equation (A1) one should follow the same path as equations (A6), (A7) and (A8)

…A1†

If the alloy system consists of three alloying elements, A, B, and C; the free energy of the system for a g phase (f.c.c. structure) consisting of A, B, and C, assuming that the system is an ideal solid solution, can be written as

…A6†

dG g ˆ ÿ G gA ‡ G gB dX g…b†B

APPENDIX A Calculation of concentration of alloying elements on stacking faults

1279

…A5†

By substituting equation (A4) in equation (A2) and obtaining the ®rst derivative with respect to X(b)B the following equation is obtained:

…A13†

where, X g…b†A ˆ 1 ÿ X g…b†B ÿ X g…b†C . In a multi-component system equation (A13) can be expressed in a general form as equation (A14) " X e…s†i ˆ 1 ‡

X X g…b†j i

X

g …b†i

#  4e 4e  ÿ1 DG gÿ ÿ DG gÿ j i exp : …A14† RT