Journal of Nuclear Materials 429 (2012) 13–18

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Journal of Nuclear Materials

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Phase decomposition and morphology characteristic in thermal aging Fe–Cr alloysunder applied strain: A phase-field simulation

Yong-Sheng Li ⇑, Hao Zhu, Lei Zhang, Xiao-Ling ChengEngineering Research Center of Materials Behavior and Design, Ministry of Education, Nanjing University of Science and Technology, Nanjing 210094, China

h i g h l i g h t s

" Effects of variation mobility and applied strain on phase decomposition of Fe–Cr alloy were studied." Rate of phase decomposition rises as aging temperature and concentration increase." Phase transformation mechanism affects the volume fraction of equilibrium phase." Elongate morphology is intensified at higher aging temperature under applied strain.

a r t i c l e i n f o

Article history:Received 21 October 2011Accepted 18 May 2012Available online 26 May 2012

0022-3115/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jnucmat.2012.05.026

⇑ Corresponding author. Tel./fax: +86 25 84315159E-mail address: ysli@njust.edu.cn (Y.-S. Li).

a b s t r a c t

The phase decomposition and morphology evolution in thermal aging Fe–Cr alloys were investigatedusing the phase field method. In the simulation, the effects of atomic mobility, applied strain, alloy con-centration and aging temperature were studied. The simulation results show that the rate of phasedecomposition is influenced by the aging temperature and the alloy concentration, the equilibrium vol-ume fractions (Ve

f ) of Cr-rich phase increases as aging temperature rises for the alloys of lower concen-tration, and the Ve

f decreases for the alloys with higher concentration. Under the applied strain, theorientation of Cr-rich phase is intensified as the aging temperature rises, and the stripe morphology isformed for the middle concentration alloys. The simulation results are helpful for understanding thephase decomposition in Fe–Cr alloys and the designing of duplex stainless steels working at hightemperature.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

As an important structural material, the Fe–Cr alloys are used forthe stainless steels and parts of the nuclear fusion reactor [1,2]. Un-der the higher service temperature, the spinodal decompositionhappens due to the existences of a miscibility gap in Fe–Cr system,in where the ferrite a decompose into Fe-rich a-phase and Cr-richa0-phase [3–5]. The phase decomposition induces the embrittle-ment at high temperature, and results in the increase of hardnessand decrease of ductility [6–8]. Soriano-Vargas et al. [6] studiedthe relationship of concentration wavelength variation and thehardness of Fe–Cr alloys aging, they found that the embrittlementis due to the formation of Cr-rich phase takes place rapidly at theearly stages of aging. Terentyev’s [7] research results indicate thatthe substantial contribution to hardening comes from edge disloca-tions shearing Cr-rich precipitates (of mean size � 3 nm [9]), due tothe higher shear modulus of the coherent body-centered cubicCr-rich precipitates, the dislocation–precipitate interaction is

ll rights reserved.

.

repulsive. So the phase decomposition of Fe–Cr alloys is an impor-tant influence factor of the mechanical properties of stainless steels.

Therefore, the phase decomposition in Fe–Cr alloys has attractedmany attentions, such as the thermodynamics [10–13], kinetics[14–17] and the relationship of morphology and mechanical prop-erties [18–20]. Bonnet et al. [20] and Pumphrey and Akhurst [12]found that the macroscopic mechanical properties can be affectedeven at the service temperature 280 �C of the cooling circuits, thechemical composition, homogenization treatment and the servicetemperature influence the mechanical properties significantly.Hyde et al. [13] studied the effect of aging temperature on the phasedecomposition of Fe–45%Cr alloy using the dynamic Ising model, inwhere the diffusion coefficient was not modeled. However, the sim-ulation presents that phase decomposition proceeds rapidly andforms the larger scales microstructure at 400 �C than that of theaging experiment. Miller and Russell [14] made comparison of therate of decomposition of the ferrite in Fe–45 wt.% Cr and Fe–45 wt.% Cr–5 wt.% Ni alloys during long-term aging at 400 �C. Theirresults showed that the kinetics of precipitation plays an importantrole in the deterioration of mechanical properties, and thecomposition also plays a key role in determining the kinetics of

14 Y.-S. Li et al. / Journal of Nuclear Materials 429 (2012) 13–18

precipitation. Ujihara and Osamura [17] analyzed the spinodaldecomposition of Fe–Cr alloys by using the Langer-Bar-on-Miller(LBM) theory, in where the composition-dependent mobility wasconsidered with the modified LBM theory, their results showed thatthe driving force and diffusion coefficient determine the rate ofspinodal decomposition. The cold-deformation on ferrite decompo-sition in duplex stainless steel was investigated by Hätestrand et al.[21], they found that the decomposition was affected by the compo-sition, temperature and the prior elastic–plastic deformation. Milleret al. [3] and Hyde et al. [22,23] investigated the spinodal decompo-sition of Fe–Cr alloys by combining experimental, theoretical andnumerical simulation. They found the time scaling regime existsfor the precipitates size, and the time exponent is given for the spin-odal decomposition. They also found that the interfaces betweenFe-rich a and Cr-rich a0 regions exhibit fractal between experimen-tal and Monte Carlo simulation, and the Cahn–Hilliard–Cook modeldo not.

In view of the above studies, the phase decomposition of Fe–Cralloys can be affected by the thermodynamics, kinetics, alloy com-position and the elastic–plastic strain. Thus, it’s necessary to studythe multiple effects to clarify the phase decomposition and themicrostructure evolution of Fe–Cr alloys. The phase field modelbased on the Cahn–Hilliard equation is effective to do this work.The aim of this paper is to investigate the effects of aging temper-ature, composition, atomic mobility and applied strain on thephase decomposition and microstructure evolution of Fe–Cr alloysby using the phase field simulation.

2. Model and methods

The total free energy of the phase decomposition in Fe–Cr alloysincludes the chemical free energy and the elastic strain energy in-duced by the composition inhomogeneous. When an externalstrain was applied to the system, the total elastic strain includesthe eigenstrain and applied strain. The total free energy F of the al-loy system can be expressed as [24]

F ¼Z

V

1Vm

Gþ 12jðrcÞ2

� �þ Eel

� �dV : ð1Þ

In where Vm is the molar volume of the alloy, j is the gradientenergy coefficient, c is the norminal composition of Cr, Eel is theelastic energy density per unit volume, and G is the molar Gibbsenergy and given by [25]

G ¼ ð1� cÞG0Fe þ cG0

Cr þ LFeCrcð1� cÞ þ RT½c ln c þ ð1� cÞ lnð1� cÞ� þ Gm;

ð2Þ

where G0Fe and G0

Cr are the energy of pure element in the unit ofJ/mol, the expressions are given by G0

Fe ¼ 1225:7þ 124:134T�23:5143T lnðTÞ � 0:00439752T2 � 5:89269� 10�8T3 þ 77358:5=T ,and G0

Cr ¼ �8856:94þ 157:48T � 26:908TlnðTÞ þ 0:00189435T2�1:47721� 10�6T3 þ 139250=T [26], R is the gas constant and T isthe absolute temperature, LFeCr is the interaction parameterbetween Fe and Cr, and the expression is LFeCr = 20500 � 9.68T(J/mol) [25], Gm is the magnetic ordering contribution to the Gibbsenergy and is expressed by

Gm ¼ RT lnðbþ 1Þf ðsÞ; ðJmol�1Þ: ð3Þ

In where b is the magnetic moment pre atom in Bohr magnetonand is given by

b ¼ 2:22ð1� cÞ � 0:008c � 0:85cð1� cÞ: ð4Þ

The function f(s) take the polynomial form

f ðsÞ¼ �0:90530s�1þ1:0�0:153s3�6:8�10�3s9�1:53�10�3s15; ðs<1Þ�0:06417s�5�2:037�10�3s�15�4:278�10�4s�25; ðs>1Þ

(;

ð5Þ

where s = T/T0, T0 is the critical temperature of magnetic orderingand given by

T0 ¼ 1043ð1� cÞ � 311:5c þ cð1� cÞ½1650þ 550ð2c � 1Þ�; ðKÞ:ð6Þ

The expressions of Eqs. (4)–(6) are refered in literature [25].The concentration gradient coefficient is expressed by [27]

j ¼ 16

r20LFeCr; ð7Þ

where r0 is the interatomic distance at stress-free state and changeswith composition by obeying Vergard’s law.

The composition evolution can be described by the Cahn–Hilliard diffusion equation [28]

@c@t¼ V2

mr � Mr dFdc

� �� �; ð8Þ

where M is the chemical mobility and given by the Darken’s equa-tion [29,30]

M ¼ 1Vm½cMFe þ ð1� cÞMCr�cð1� cÞ; ð9Þ

where MFe and MCr are the atomic mobility of Fe and Cr, respec-tively. They are related to the diffusivity through Einstein’s relationMi = Di/RT, where i denotes the element Fe or Cr, Di is the diffusioncoefficient and chosen as [31] DFe ¼ 1:0� 10�4 exp � 294kJ=mol

RT

� and

DCr ¼ 2:0� 10�5 exp � 308kJ=molRT

� ðm2=sÞ.

2.1. Elastic stress

To introduce the applied strain into the alloy system, the inho-mogeneous elastic modulus tensor is considered, i.e. the elasticmodulus of precipitates and the matrix are different, which canbe written as

Cijkl ¼ C0ijkl þ DCijklDc; ð10Þ

where Dc = c � c0, c0 is the average composition,C0

ijkl ¼ kCPijkl þ ð1� kÞCM

ijkl is the average modulus with k the volumefraction of the precipitates, CM

ijkl and CPijkl are the elastic modulus

tensors of the matrix phase and precipitates, respectively,DCijkl ¼ CP

ijkl � CMijkl.

The total elastic strain of the system including the applied straincan be written as

eelij ¼ ea

ij þ eij � e0ij; ð11Þ

where eaij is the applied strain, eij is the internal strain, e0

ij is theeigenstrain caused by the compositional inhomogeneity and givenby

e0ij ¼ e0dijDc; ð12Þ

where e0 = 1/a(da/dc) is the composition expansion coefficient ofthe lattice parameter and dij is the Kronecker-delta function.

Then the local elastic stress can be given by Hook’s law,

relij ¼ ðC

0ijkl þ DCijklDcÞðea

kl þ ekl � e0klÞ: ð13Þ

By using the relationship of displacement ui and internal strainekl

ekl ¼12

@uk

@xlþ @ul

@xk

� �; ð14Þ

where ui denotes the ith component of the displacement. The inter-nal strain can be obtained by solving the mechanical equilibriumequation

Fig. 1. Phase decomposition morphology with the elevated aging temperature under applied strain eayy ¼ 0:015, t� = 700. (a–c) c0 = 0.4; (d–f) c0 = 0.5. (a) T = 640 K, (b)

T = 670 K, (c) T = 700 K, (d) T = 640 K, (e) T = 670 K, (f) T = 700 K.

Y.-S. Li et al. / Journal of Nuclear Materials 429 (2012) 13–18 15

@relij

@xj¼ 0: ð15Þ

By substituting Eqs. (12)–(14) into Eq. (15), we have

C0ijkl

@2uk

@xj@xlþ DCijkl

@

@xjDc

@uk

@xl

� �

¼ C0ijkle0dkl

hþ 2DCijkle0dklDc � DCijklea

kl

@Dc@xj

: ð16Þ

Eq. (16) is a nonlinearly mechanical equilibrium equation and itsanalytic solution is obtained by the first-order approximation[32]. Solving Eq. (16) in Fourier space, the displacement and theinternal strain can be given. Then, the elastic strain energy densityper unit volume can be calculated by

Eel ¼12

Cijkleelij e

elkl ¼

12

Cijklðeaij þ eij � e0

ijÞðeakl þ ekl � e0

klÞ: ð17Þ

Fig. 2. Morphology of Fe–Cr alloy aging at 700 K under applied strain eayy ¼ 0:015, t�

2.2. Numerical calculation

By substituting Eq. (1) into Eq. (8), the composition evolutionequation is given by

@c@t¼ Vmr � Mr dG

dc� jðrÞ2c þ Vm

dEel

dc

� �� �: ð18Þ

The dimensionless form of Eq. (18) can be written as

@cðr�; t�Þ@t�

¼ r� � M�r� dG�

dc� j�ðr�Þ2c þ dE�el

dc

� �� �: ð19Þ

In where r� = r/l, t� = tD/l2, M⁄ = VmRTcM/D, G� = G/RTc, j⁄ = j/RTcl

2, E�el ¼ VmEel=RTc , l is the grid length and chosen as a0, a0 isthe average lattice parameter of Fe and Cr for c0 = 0.5,D = 10�27m2s�1 is normalization factor of the diffusion coefficient,Tc = 900 K is the critical temperature of spinodal decomposition of

= 140. (a) c0 = 0.35, (b) c0 = 0.4, (c) c0 = 0.46, (d) c0 = 0.5, (e) c0 = 0.54, (f) c0 = 0.6.

0 100 200 300 400 500 600

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time Step (X200)

Volu

me

Frac

tion

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

Time Step (X200)

Volu

me

Frac

tion

(a)

(b)

Fig. 3. The volume fraction variation of a0 phase as a function of time with andwithout applied strain. (a) c0 = 0.32, T = 660 K, (b) c0 = 0.5, T = 700 K.

16 Y.-S. Li et al. / Journal of Nuclear Materials 429 (2012) 13–18

the alloy [3], the lattice parameter are aFe = 0.2866 nm andaCr = 0.2882 nm [33], so the molar volume is Vm = 1.4 �10�5 m3 mol�1. The grid size are chosen as Dx = Dy = l, thus thedimensionless grid size are Dx� = Dy� = 1.0, the simulation cell sizeis adopted as 128 Dx� � 128 Dy�. A thermal fluctuation[�0.002,0.002] is introduced into the initial composition in thesimulation. The elastic constants of Fe are chosen as CFe

11 ¼ 205,CFe

12 ¼ 129, CFe44 ¼ 109 GPa at 672 K [34] and element Cr are

CCr11 ¼ 365, CCr

12 ¼ 115 and CCr44 ¼ 96 GPa at 650 K [35]. The Eq. (19)

is solved by using the semi-implicit Fourier spectrum method[36] with the time step Dt� = 0.007.

10 20 30 40 500

0.2

0.4

0.6

0.8

1

Com

posi

tion

Pos

Fig. 4. Composition evolution of a0 phase as a functio

3. Results and discussion

3.1. Effects of applied strain on the morphology

Applied strain induces the additional elastic energy, in order toreduce the elastic energy, the precipitates will arrange along theelastic soft direction, so the morphology is dominated considerablyby the applied strain. At the same time, the atomic mobility relatedwith temperature will affect the morphology during aging process.Fig. 1 shows the Fe–Cr alloy phase decomposition at different agingtemperatures under the applied strain ea

yy ¼ 0:015 along y axis, inwhere the Cr-rich a0 and the Fe-rich a are represented by the whiteand black, respectively. The aging time is t� = n � Dt� = 700, wheren is the calculation time step number. When the alloy compositionis c0 = 0.4, the a0 phase presents the isolated particle, and it elon-gates along the level direction as the aging temperature increasesfrom T = 640 K to 700 K, as shown in Fig. 1a–c. It also can be seenthat the particles growth and coarsening are faster at higher agingtemperature for the same aging time.

When the composition increases to c0 = 0.5, the modulatedmorphology is presented, see Fig. 1d, in which the orientation isnot obvious at the lower aging temperature 640 K even if underthe applied strain ea

yy ¼ 0:015. However, the elongated stripe shapea0 morphology is formed as the aging temperature increases to700 K, see Fig. 1f, the orientation of the soft a0 phase is perpendic-ular to the direction of applied strain.

The above results show that the atomic mobility increases asaging temperature rises, the influence of applied strain on the mor-phology is intensified and the phase transformation is faster. Thus,the rate of phase decomposition and morphology under the ap-plied strain are influenced by the aging temperature. The simu-lated aging morphology and particles size for the c0 = 0.4 werecompared with the experimental results given in literature [6],the isolated particles and it’s size in Fig. 1b are similar to the exper-imental TEM image.

Fig. 2 shows the morphology evolution of Fe–Cr alloy with dif-ferent compositions under the applied strain ea

yy ¼ 0:015, the agingtemperature is 700 K and the aging time is t⁄=140. It can be seenthat the morphology of a0 phase changes from the isolated ellipseof lower concentration to the stripe shape of higher concentration,as shown in Fig. 2a–d. When the composition increases to c0 = 0.6,the a0 phase is interconnected and the a phase shows the isolatedellipse, see Fig. 2f. The results showed that the continued stripstructure of a0 phase can be formed under the applied strain forthe middle concentration alloys in Fe–Cr alloys, the single-axismorphology may influences the mechanical properties of the stain-less steels at higher service temperature.

60 70 80 90 100ition

t*=2000t*=3000t*=5000t*=8000t*=14000t*=24000

n of time for alloy with c0 = 0.22 aging at 600 K.

0 200 400 600 8000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time Step (X200)

Volu

me

Frac

tion

T=700KT=670KT=640K

0 200 400 600 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time Step (X200)

Voul

ume

Frac

tion

T=700KT=670KT=640K

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

Time Step (X200)

Volu

me

Frac

tion

T=700KT=670KT=640K

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

Time Step (X200)

Volu

me

Frac

tion

T=700KT=670KT=640K

(a) (b)

(c) (d)

Fig. 5. The volume fraction variation of a0 phase as a function of time for different aging temperatures and concentrations. (a) c0 = 0.35, (b) c0 = 0.46, (c) c0 = 0.5, (d) c0 = 0.6.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

c0=0.35c0=0.46c0=0.60

Mob

ility

Temperature (K)640 650 660 670 680 690 700

640 650 660 670 680 690 7000.2

0.3

0.4

0.5

0.6

0.7 c0=0.35c0=0.46c0=0.60

Volu

me

fract

ion

Temperature (K)

(a)

(b)

Fig. 6. The change of chemical mobility (a) and volume fraction of a0 phase (b) as afunction of aging temperature for different alloy concentrations.

Y.-S. Li et al. / Journal of Nuclear Materials 429 (2012) 13–18 17

As referred above, the applied strain influences the morphologyof a0 phase especially at higher aging temperature. Furthermore,we want to know how about the effects of applied strain on therate of phase decomposition. In order to clarify this question, thevariations of a0 phase volume fractions (Vfs) as a function of timewith and without the applied strain were plotted in Fig. 3. It’sshown that the applied strain ea

yy ¼ 0:015 has almost no effect onthe initial phase decomposition for alloy with c0 = 0.32 aging at660 K, see Fig. 3a. The variations of Vfs are the same with and with-out applied strain for alloy with c0 = 0.5 aging at 700 K, see Fig. 3b.Besides, the effects of applied strain on the Vf for other concentra-tions and aging temperatures were also studied, all the resultsshowed that the applied strain has no obvious influence on the rateof Fe–Cr alloy phase decomposition. This can be attributed to thevery small lattice mismatch 0.0056 of the Fe–Cr system, wherethe additional energy induced by the applied strain is too smallto change the rate of phase transformation.

3.2. Effects of aging temperature and concentration on phasedecomposition

In this section, the effects of aging temperature and concentra-tion on the Fe–Cr alloy phase decomposition were investigated. Ashave referred in the literatures [37–40], the phase transformationmechanisms of Fe–Cr alloy change from the nucleation and growthto spinodal decomposition as the concentration increases.

Firstly, the phase transformation mechanism was investigatedfor the alloy with c0 = 0.22 aging at 600 K. In order to trigger thephase separation, a larger initial thermal fluctuation [�0.1,0.1] isadded and the circle time step number is n = 6000, the time stepis Dt� = 0.2 in this calculation. Fig. 4 shows the composition evolu-tion of a0 phase as a function of time. It can be seen that the com-position has a large fluctuation at t� = 2000, it approaches to theequilibrium value at t� = 5000. At the same time, the compositionaround the a0 phase particles decreases. Then the composition

18 Y.-S. Li et al. / Journal of Nuclear Materials 429 (2012) 13–18

and particle size of a0 phase increase gradually. At t� = 24000, thecomposition of a0 phase and a phase reaches the equilibrium value.Since the phase decomposition needs a large enough thermal fluc-tuation, the phase transformation mechanism is nucleation andgrowth for alloy with c0 = 0.22. On the other hand, the compositionevolution is step by step, so the phase transformation is a non-clas-sical nucleation mechanism, which is also demonstrated by Novyet al. [38] in Fe–Cr alloy with c0 = 0.2. As the concentration in-creases, such as c0 = 0.35, the phase decomposition mechanism willtransform to the spinodal decomposition.

Fig. 5 shows the change of a0 phase Vf as a function of time atdifferent aging temperatures with the applied strain ea

yy ¼ 0:015.It can be seen that the Vf reaches the equilibrium value firstly atthe higher aging temperature 700 K for the alloys with c0 = 0.35,0.46, 0.5 and 0.6, as the aging temperature decreases, the time islonger for Vf reaching equilibrium. This means that the diffusiondominates on the phase transformation, and the aging temperatureinfluences the rate of phase decomposition.

The effects of concentration on the phase decomposition can bedetected from Fig. 5, in which the aging temperature is 640 K, theVf reaches the equilibrium value at t� = 700, 490 and 700 for alloywith c0 = 0.35, 0.46 and 0.6, respectively. So the phase decomposi-tion is faster for Fe–Cr alloys with middle concentration. Further-more, the relative magnitude of equilibrium volume fraction Ve

f

for different aging temperatures also changes with the composi-tion. As shown in Fig. 5a, the Ve

f ¼ 0:33 at 640 K is larger than thatof Ve

f ¼ 0:285 at 700 K for alloy with c0 = 0.35. When the concentra-tion is c0 = 0.46, the Ve

f s are almost equal at different temperatures,see Fig. 5b. However, the Ve

f at 640 K is less than that of 700 K foralloy with c0 = 0.6. Thus, the rate of phase decomposition and theVe

f of a0 phase are influenced by the aging temperature and theconcentration.

Fig. 6a shows the variation of chemical mobility with the agingtemperature. It can be seen that the mobility increases as the agingtemperature rises, and the higher the alloy concentration, the lar-ger the mobility. For the lower concentration alloy, such asc0 = 0.35, it locates at the transition regions of nucleation andgrowth to spinodal decomposition [37–41]. Although the mobilityis larger at 700 K, the supercooling of phase transformation is lessthan that of 640 K. So the driving force of phase transformation isless, the precipitates decrease at higher temperature and results inthe decrease of Vf, as shown in Fig. 6b labeled with the square.However, as the alloy concentration increases, the phase transfor-mation mechanism is full spinodal decomposition, the infinitesi-mal composition fluctuation will induce the phasedecomposition, i.e. there is no energy barrier for the spinodaldecomposition. The larger mobility of high temperature results inthe fast composition clustering. When the concentration is greaterthan c0 = 0.46, the a0 phase is interconnected during the phasedecomposition, the Ve

f of a0 phase increases as temperature rises,see Figs. 6b and 5c–d.

4. Conclusions

In this paper, the phase field simulation is performed to inves-tigate the influence of atomic mobility, applied strain, aging tem-perature and concentration on the phase decomposition andmorphology in the thermal aging Fe–Cr alloys. The simulation re-sults show that the rate of phase decomposition is increased withthe elevated temperature due to the increased atomic mobility.When the concentration is less than 0.46, the relative magnitudeof equilibrium volume fraction of Cr-rich phase decreases as the

aging temperature increases, and the inverse variation is presentedwhen the concentration is greater than 0.46. The results also showthat the morphology orientation is intensified with the elevatedaging temperature under the applied strain, the morphology ofCr-rich phase changes from the isolated ellipse to strip shape asthe concentration increases, and the orientation is perpendicularto the applied strain direction.

Acknowledgments

This work was funded by the National Natural Science Founda-tion of China (No. 5100 1062), Jiangsu Provincial Natural ScienceFoundation (No. BK2011710) and NUST Research Funding (No.2011YBXM160).

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