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Materials Science and Engineering A 528 (2011) 8628– 8634

Contents lists available at SciVerse ScienceDirect

Materials Science and Engineering A

journa l h o me pa ge: www.elsev ier .com/ locate /msea

hase field simulation of precipitates morphology with dislocations underpplied stress

ong-sheng Lia,b,∗, Yan-zhou Yua,b, Xiao-ling Chenga,b, Guang Chena,b

Engineering Research Center of Materials Behavior and Design, Ministry of Education, Nanjing 210094, ChinaSchool of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

r t i c l e i n f o

rticle history:eceived 17 December 2010eceived in revised form 9 August 2011ccepted 16 August 2011vailable online 22 August 2011

a b s t r a c t

A phase field dynamic model was developed and used to investigate the effects of dislocations and appliedstrain on the precipitation behavior and microstructure evolution of model binary alloys. The simulationsshow that the local microstructure depends not only on the relative magnitude of the dislocation stressand the stress induced by the applied strain, but also on the composition and magnitude of the stress. Itsalso shown that the applied strain makes the phase decomposition quickly. The results suggest that the

eywords:recipitationicrostructureislocationpplied stress

microstructure of an alloy and its evolution may be controlled by finding suitable combination betweendislocation, applied strain and composition, and the theoretical calculations are helpful in predictingwhat those combinations should be.

© 2011 Elsevier B.V. All rights reserved.

hase field

. Introduction

Applied stress and dislocation stress affect the morphologyf alloy during the solid-state phase transformation. An appliedtress can introduce an additional lattice mismatch strain whenhe elastic constants of the matrix and the precipitates are differ-nt; it also induces the coupling elastic strain energy during thetructure transformation. The applied stress can also modify theorphologies of the precipitate particles, e.g. their size, shape, vol-

me fraction, orientation with respect to the matrix, and phaseransformation kinetics, thus affecting the mechanical and physicalroperties of a material [1–7]. On the other hand, the dislocationtress field can change the velocity of migrating atoms and theirection in the diffusion phase transformation, influencing in thisay the local precipitates morphology and the phase transforma-

ion velocity [8–11], which in turn results in a change of the localicrostructure. In addition to the applied stress and dislocation

tress, the coherency elastic stress arising from the crystal latticeismatch between the matrix and the precipitates is also impor-

ant, and it affects the solid-state transformation as well [12,13].or predicting and controlling the properties and morphology ofaterials it is thus crucial to understand how the applied stress, dis-

∗ Corresponding author at: School of Materials Science and Engineering, Nanjingniversity of Science and Technology, Nanjing 210094, China.el.: +86 25 8431 5159; fax: +86 25 8431 5159.

E-mail address: ysli@mail.njust.edu.cn (Y.-s. Li).

921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2011.08.024

locations and coherency stress affect the microstructures’ evolutionduring a solid-state phase transformation [14–17].

Numerous experimental and theoretical studies have beendevoted to investigate this issue [1–13]. For example, Miyazakret al. [3] investigated the morphological changes of �′ particles inNi–15 at.%Al alloy single crystals due to annealing at 1023 K undertensile and compressive loads in a [0 0 l] cube direction. They foundthat both rods and plates of �′ are aligned parallel to the tensileaxis [0 0 l] and perpendicular to the compressive axis [0 0 l]. On theother hand, Fährmann et al. [7] studied the effect of pre-strain andthe development of rafting during aging. They found that the pre-strain paths modify the initial structure of �/�′ interfaces and thatthe local state of stress contributes the driving force for rafting. The-oretical studies that employed the phase field simulation techniquehave been used to investigate the effects of applied stress and dislo-cation stress on a solid-phase transformation [4–11]. For example,Li and Chen [4] investigated the shape evolution and splitting pat-tern of coherent particles under applied stresses. They found thatthe elongation direction of the precipitates was influenced by theapplied strain direction, the relative magnitude of the elastic con-stant of the precipitates and matrix, and the sign of the latticemismatch. Gururajan and Abinandanan [5] studied the precipitatesrafting under the uniaxial stress, and their results showed that thepurely elastic stress-driven rafting is possible, the rafting is more

pronounced for the soft precipitates, and the sign of the appliedstress is the same as that of the misfit. Finel et al. [6] analyzed themicrostructure evolution in the presence of a lattice misfit and withinhomogeneous elastic constants; they found that the external load

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Y.-s. Li et al. / Materials Science an

long an axis makes the microstructure anisotropic, and that theituation qualitatively differs depending on the sign of the appliedtress.

The dislocation effects on the precipitation behavior and mor-hology have been studied by Hu and Chen [10] with Mura’s [18]islocation eigenstrain in Fourier space with periodic boundaryondition. Thus, they found that coherent nucleation may becomearrierless under the influence of the local elastic field of a dislo-ation. Li et al. [11] investigated the effects of dislocations on thee–Cr alloy spinodal decomposition using Stroh’s [19,20] disloca-ion formula, and their results showed that dislocations facilitatehe phase decomposition and that special morphologies appearednduced by the tilt grain boundary. Finally, He [21] and Chen [22]tudied the spinodal morphology of thin film with periodic dislo-ations; their results showed that the dislocations change the localicrostructure pattern.All the studies above have focused on the effect that either

pplied stress or dislocations stress introduce in a solid-stateransformation. The combined effect of applied elastic strain andislocations on the morphology is not very clearly yet, and that isn interested question. In this paper, we will study this combinedffect by investigating the precipitation behavior and microstruc-ure of model binary alloys with dislocations under applied strain.he effects of the relative magnitude of applied stress and dislo-ation stress on the local morphology and phase transformationelocity were also investigated.

. Methodology

In this section we describe the model that has been used tonvestigate how dislocation and applied stress affect the morphol-gy of alloy during a solid-state transformation.

.1. The phase field model

The microstructure evolution of a binary alloy, A–B, during pre-ipitation can be described by the solute composition c(x, t) at anyoint x at time t, and it is governed by the Cahn–Hilliard diffusionquation [23,24]

∂c

∂t= ∇ ·

[M · ∇

(ıF

ıc

)], (1)

here M is the chemical mobility and c is the atom fraction of thelement B. F in the equation above is the total free energy of theimulated system, and it is given by the expression

=∫

V

[f (c) + 1

2k(∇c)2 + Eel

]dV , (2)

here f (c) is the local chemical free energy density per unit volume,/2k(∇ c)2 represents the concentration gradient energy per unitolume, and Eel is the elastic energy density per unit volume.

The chemical free energy density per unit volume of A–B for theegular solution approximation is given by

(c) = (1 − c)G0A + cG0

B + ˝c(1 − c) + RT[c ln c + (1 − c) ln(1 − c)]Vm

,

(3

here G0A and G0

B are the standard molar Gibbs free energies ofure A and B, respectively, is the interaction parameter, chosens = 18 kJ/mol, R is gas constant, T is the absolute temperature,nd Vm denotes the molar volume of the alloy. G0

A = G0B = 0 was

dopted as the reference energy level for the Gibbs free energy.The concentration gradient coefficient is expressed as [25,26]

= 1Vm

16

r02˝, (4)

neering A 528 (2011) 8628– 8634 8629

where r0 is the interatomic distance at stress-free state and changeswith composition by simply obeying Vergard’s law. The mobility Mis assumed to be a constant for simplicity.

The phase field equation results from substituting Eq. (2) intoEq. (1), and the final result is given by the expression

∂c

∂t= M∇2

[ıf (c)

ıc− k∇2c + ıEel

ıc

]. (5)

2.2. Elastic stress

The elastic energy in Eq. (5) includes the energy induced by theeigenstrain, applied and dislocation strain. To introduce the appliedstrain into the total elastic energy, the inhomogeneous elastic mod-ulus tensor is considered, i.e. the elastic modulus of precipitatesand the matrix are different. The local elastic modulus tensor canbe represented as follows:

Cijkl = C0ijkl + �Cijkl�c, (6)

where �c = c − c0, c0 is the average composition at the zero stressreference, C0

ijkl= �CP

ijkl+ (1 − �)CM

ijklis the average modulus with

� the volume fraction of the precipitates, and CMijkl

and CPijkl

arethe elastic modulus tensors of the matrix phase and precipitates,respectively, �Cijkl = CP

ijkl− CM

ijkl.

The elastic strain of the system including the applied strain anddislocation can be given as

εelij = εa

ij + εij − ε0ij − εd

ij , (7)

where εaij

is the applied strain, εij is the internal strain, ε0ij

is theeigenstrain caused by the compositional inhomogeneity and isgiven by

ε0ij = ε0ıij�c, (8)

where ε0 = 1/a(da/dc) is the composition expansion coefficient ofthe lattice parameter and ıij is the Kronecker-delta function. Thedislocation eigenstrain εd

ijin Eq. (7) for an edge dislocation with

Burgers vector b = (b1, 0, 0) can be expressed as [18]

εd21 = 1

2b1ı(x2)H(−x1), (9)

where ı(x2) is Dirac’s delta function and H(− x1) is the Heavisidestep function, they each has the property

ı(x − x0) ={

0 (x /= x0)+∞ (x = x0)

, (10)

H(−x1) ={

1 x1 < 00 x1 > 0

. (11)

The other components of the eigenstrain εdij

are zero. In the cal-culation, the Burgers vector of the dislocation is expressed by theGaussian function given in the literature [10].

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

�elij = (C0

ijkl + �Cijkl�c)(εaij + εij − ε0

ij − εdij). (12)

By using the relationship of displacement ui and internal strain εkl,

εkl = 12

{∂uk

∂xl+ ∂ul

∂xk

}, (13)

where ui is used to denote the ith component of the displacement.The internal strain can be obtained by solving the mechanical equi-

librium equation

∂�elij

∂xj= 0. (14)

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630 Y.-s. Li et al. / Materials Science an

ubstituting Eqs. (8) and (12) into the mechanical equilibriumquation (14), we have

0ijkl

∂2uk

∂xj∂xl+ �Cijkl

∂

∂xj

(�c

∂uk

∂xl

)

= [C0ijklε0ıkl + 2�Cijklε0ıkl�c − �Cijklε

akl + �Cijklε

dkl]

∂�c

∂xj

+(C0ijkl + �Cijkl�c)

∂εdkl

∂xj(15)

Eq. (15) is a nonlinearily mechanical equilibrium equation andts analytic solution is obtained by the first-order approximation2,4], i.e. the uk in the second item of the left side is looked as the0k

and this item is moved to the right side of Eq. (15). The dis-lacement u0

kcan be solved firstly by the zeroth-order approach

ith �Cijkl = 0 in Eq. (15). Using the similar approach, the high-rder solution of Eq. (15) can be obtained. Solving Eq. (15) in Fourierpace, the displacement is given by

˜k = −iGikkj

[(C0

ijmnε0ımn + 2�Cijmnε0ımn�c − �Cijmnεamn

+�Cijmnεdmn)�c + (C0

ijmn + �Cijmn�c)εdmn

−�Cijmn

(�c

∂u0m

∂xn

)k

], (16)

here G−1ij

(k) = C0imlj

kmkl , k is reciprocal vector in Fourier space, ki

s the component of ith, i = √−1, uk, �c, εdmn and (�c(∂u0

m/∂xn))kepresents the Fourier transformation of uk, �c, εd

mn andc(∂u0

m/∂xn), respectively. The corresponding strain εkl in Fourierpace is

˜kl = i

2[ukkl + ulkk]. (17)

Thus, the elastic strain energy density per unit volume can bealculated by

el = 12

Cijklεelij εel

kl = 12

Cijkl(εaij + εij − ε0

ij − εdij)(ε

akl + εkl − ε0

kl − εdkl).

(18)

.3. Numerical calculations

Eq. (5) is numerically solved using a Fourier spectral techniquender the periodic boundary conditions [27]. To do this, we use theimensionless parameters x* = x/l, t* = tM�Vm/l2, f* = f/�, k* = k/�l2,∗el

= Eel/�, where � is the shear modulus of element B, l is theength scale and chosen as 10|b|, |b| is the magnitude of Burgersector b = a0/2 [1 0 0] on the (0 1 0) slip plane of the bcc structure,nd a0 is the lattice constant of the alloy with the length 0.3 nm. Therid size is chosen as �x1 = �x2 = l, thus the dimensionless grid sizes �x∗

1 = 1.0 and �x∗2 = 1.0. The composition expansion coefficient

f the lattice parameter is ε0 = 0.03, and the gradient coefficients k* = 1.0. The elastic constants of element A are CA

11 = 230, CA12 =

30, CA44 = 170, and element B are CB

11 = 350, CB12 = 67 and CB

44 =00 GPa.

. Results and discussion

The initial state of the alloy system is a homogeneous solid solu-

ion with a small random fluctuation, [−0.001, 0.001], around theverage composition. The size of the simulation cell is 128�x∗

1 ×28�x∗

2, and the time step is �t∗ = 0.02. The precipitates and matrixere represented by the B-rich ˛′ phase and the A-rich phase.

ineering A 528 (2011) 8628– 8634

Since the shear modulus of element B is less than that of element A,the precipitation is softer than the matrix [4,5], so the applied strainis along the y-axis (x2-axis) and produces the elongated precipitatesalong the x-axis (x1-axis).

3.1. Microstructure with one dislocation and applied strain

Fig. 1 shows precipitation process of the ˛′ phase for the alloywith c0 = 0.45 aging at T = 560 K, and an edge dislocation with Burg-ers vector b = a0/2 [1 0 0] is located at the position (64, 64) in thesimulation cell. There is no applied strain in Fig. 1a–d, whereas thestrain εa

yy = 0.01 is applied in Fig. 1e–h.From Fig. 1a, b, e and f it is clear that, independently of the

applied strain, the precipitation morphology of ˛′ is similar. The˛′ first nucleates at the tensile stress region of the edge disloca-tion and it shows the circle waves microstructure, see Fig. 1b andf. Thus, the effect of the applied strain on ˛′ precipitation and mor-phology is not obvious at the initial stage. Instead, the precipitationand morphology of ˛′ are influenced by the dislocation stress at thisinitial stage.

As aging progress the situation changes. Without applied strain,the ˛′ distributes itself randomly far from the dislocation, see Fig. 1cand d, whereas at the dislocation regions it still shows the circu-lar shape. On the other hand, when the strain εa

yy = 0.01 is applied,both far and near to the dislocation region, an alignment in thex-direction takes place, see Fig. 1g. Finally, after nucleation andgrowth, ˛′ coarsens in bands, see Fig. 1h. Thus, it can be concludedthat the applied strain weakens the effect introduced by dislocationon the local morphology.

Further insights are gained by plotting the composition evolu-tion along x∗

1 = 64. This was done in Fig. 2, it can be seen from Fig. 2a,b, e and f, the composition evolution with and without appliedstrain is similar at the initial stage. As aging progresses, the compo-sition is more regular under the applied strain, but whether a strainis applied or not, the composition reaches an equilibrium valuepractically simultaneously, Fig. 2c, d, g and h. This indicates that theapplied strain has no effect on the velocity of phase decompositionfor the alloy with c0 = 0.45.

The combined effect of dislocation and applied strain is inves-tigated further in Fig. 3. Here we show the alloy with c0 = 0.43,T = 560 K, an edge dislocation at (64, 64) with Burgers vector b = a0/2[1 0 0], and three applied strains εa

yy : 0.002, 0.005 and 0.01. As theapplied strain increases the alignment of ’ becomes more evi-dent. However, the effect of dislocation on the local morphologyis damped as the applied strain increases. Its also can be knownthat no matter what the magnitude of the applied strain is, ’ stillnucleates at the dislocation, which is indicated by the white arrowsin Fig. 3.

The reason why ’ still grows at the dislocation core region, nomatter the strain’s magnitude, is that at the core region the dislo-cation stress is larger than the stress induced by the applied strain.This is clearly seen in Fig. 4, Fig. 4a shows the contour of hydrostaticstress of an edge dislocation calculated by the average modulus ofprecipitates and the matrix for c0 = 0.5, the lines denote the stressdistribution and the numbers with plus sign the magnitude of thecorresponding stress; Fig. 4b shows the applied strain and inducedstress relation. In the simulation system, the stress induced by theapplied strain is the same everywhere, however, as seen in Fig. 4a,the dislocation stress decreases gradually from the center to the cellboundary. This creates situations in which the dislocation stress isdifferent to the stress induced by the applied strain. Indeed, theapplied strain 0.02 corresponds to an induced stress of 7.8; as it

is seen in Fig. 4a, this stress is larger than the dislocation stress farfrom the dislocation center (where the absolute value of dislocationstress can be 3.35 and 5.58), but it is smaller than the dislocationstress at the core regions (where the absolute value of dislocation

Y.-s. Li et al. / Materials Science and Engineering A 528 (2011) 8628– 8634 8631

Fig. 1. Microstructure of the alloy aging with an edge dislocation located at (64, 64) with the Burgers vector b = a0/2 [1 0 0], c0 = 0.45, T = 560 K. (a)–(d) Without applied strain;(e)–(h) with the applied strain εa

yy = 0.01.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1Composition

Position0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Composition

Position0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Composition

Position

(e) t*=200 (f) t*=800 (g) t*=1000 (h) t*=3000

(a) t*=200 (b) t*=800 (c) t*=1000 (d) t*=3000

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Composition

Position

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Composition

Position0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Composition

Position0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Composition

Position0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Composition

Position

Fig. 2. Composition evolution of ˛′ phase along x∗1 = 64. (a)–(d) Correspond to Fig. 1(a)–(d); (e)–(h) correspond to Fig. 1(e)–(h).

F disloc(

sibntd

ig. 3. Microstructure of the alloy aging with different applied strains and an edge

a) εayy = 0.002; (b) εa

yy = 0.005; (c) εayy = 0.01.

tress can be 18.3). As a consequence, the applied strain will dom-nate the precipitates arrangement far from the dislocation center,

ut at the core regions the dislocation stress is the one that domi-ates. Therefore, it can be concluded that the relative magnitude ofhe dislocation stress and the stress induced by the applied strainetermines the precipitation morphology of ’.

ation located at (64, 64) with the Burgers vector b = a0/2 [1 0 0], T = 560 K, t* = 3000.

3.2. Microstructure evolution with dislocation wall and appliedstrain

The observations of the previous can be taken one step furtherby investigating the case where the dislocation density is very high,such as the dislocation wall. In particular, we consider the case of

8632 Y.-s. Li et al. / Materials Science and Eng

0.000 0.005 0.010 0.015 0.020 0.025

0

2

4

6

8

10

Stre

ss

Applied Strain

b

-18.3-10.1

-7.82

-5.58

-3.35

3.35

5.58

7.8210.1

18.3

Position

Position

-1.12-1.12

250 255 260 265250

255

260

265

+1.12

a

Fig. 4. The contour of the hydrostatic pressure of an edge dislocation (a) and theapplied strain and stress relation (b) of the alloy system.

Fig. 5. Microstructure of the alloy aging with a dislocation wall along x

ineering A 528 (2011) 8628– 8634

an alloy with c0 = 0.45, T = 560 K under the applied strain εayy = 0.02

and εaxx = 0.02, and with a dislocation wall formed by the edge

dislocation with Burgers vector b = a0/2 [1 0 0] along x∗1 = 64. The

distance of the dislocation is d = 10.Fig. 5a–d shows the microstructure evolution when the applied

strain is εayy = 0.02. As with a single dislocation, the dislocation wall

also determines the initial morphology of ˛′. As aging progresses,at regions far from the dislocation wall the alignment of ˛′ is domi-nated by the applied stress, whereas at regions near the dislocationwall ˛′ still shows the regular elliptical particle shape, see Fig. 5b–d.The regular particle is a consequence of the fact that in those regionsthe stress induced by the dislocation wall is larger than the stressinduced by the applied stress. The same kind of alignment takesplace if one applies the strain in a different direction, i.e. εa

xx = 0.02,see Fig. 5e–h.

These results suggest that it might be possible to control themicrostructure on an alloy by suitable combination of applied stressand dislocations.

3.3. Effects of applied strain on phase decomposition

It is known that the dislocation stress makes the local phasedecomposition faster [11]. The question is what is the effect ofapplied stress on the phase decomposition and how both appliedstress and dislocation stress affect local phase decomposition.These issues are investigated in this section.

Fig. 6 shows the morphology evolution of an alloy with c0 = 0.22aging at 540 K. In Fig. 6a–c, an applied strain of εa

yy = 0.02 is consid-ered; in Fig. 6d–f, an edge dislocation is considered at the center ofthe cell under applied strain εa

yy = 0.02; finally, only an edge dislo-cation at the center of the cell is considered in Fig. 6g–i. The type ofthe dislocation in Fig. 6 is the same as that of Fig. 1. As to the mor-phology of the precipitates, it is seen that the precipitates showthe elliptical shape under the applied strain in Fig. 6a–f, ˛′ phase iselongated compared with that of in Fig. 6g–i.

The variation of ˛′ volume fraction (Vf) with time is shown inFig. 7. At the initial stage, the Vf with dislocation increases fasterbefore t* = 2800, then the Vf with applied strain is larger than that ofwithout applied strain until t* = 4000. At t* = 3500, Vf reaches a max-

imum and then declines very fast. This fast declining is due to thefact that the applied strain induces more ˛′ phase nucleus at the ini-tial stage, but as aging progresses, the metastable nucleus dissolve,and thus Vf decreases sharply. Vf stabilizes itself after t* = 5000. On

∗1 = 64. c0 = 0.45, T = 560 K. (a)–(d) εa

yy = 0.02; (e)–(h) εaxx = 0.02.

Y.-s. Li et al. / Materials Science and Engineering A 528 (2011) 8628– 8634 8633

Fig. 6. The alloy with c0 = 0.22 aging at 540 K. (a)–(c) With applied strain εayy = 0.02; (d)–(f) with an edge dislocation under applied strain εa

yy = 0.02; (g)–(i) with an edgedislocation.

50 10 15 20 25 30

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Vol

ume

Frac

tion

Time Step(X200)

ayy=0.02ayy=0.0ayy

=0.02

F ′

5c

ts

ddatiadTFa

50 10 15 20 25

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Vol

ume

Frac

tion

Time Step (X200)

ayy

=0.02ayy

=0.0

a

50 10 15 20 25

0.0

0.1

0.2

0.3

0.4

0.5

Vol

ume

Frac

tion

ayy=0.02ayy=0.0

b

ig. 7. The volume fraction of ˛ phase as a function of time for c0 = 0.22 aging at40 K. Square: denotes a situation where an edge dislocation and applied strainoexist; circle: only a dislocation exists; triangle: only applied strain exists.

he other hand, in the absence of an applied strain Vf increaseslowly, and reaches an equilibrium value at about t* = 5000.

To demonstrate the effects of the applied strain on the phaseecomposition, the variation of ˛′ volume fraction, with an edgeislocation in the cell center and aging at 540 K, are plotted in Fig. 8and b for the alloys with composition c0 = 0.3 and 0.45, respec-ively. Fig. 8a shows that at the initial stage the volume fractions higher under applied strain εa

yy = 0.02 than that is without thepplied strain. As aging progresses, Vf reaches a maximum and then

eclines fast to an equilibrium value when the applied strain exist.he declining however is not as sharp as it was when c0 = 0.22 (seeig. 7); this is because the number of metastable nucleus decreasess the concentration increases. When the alloy concentration is

Time Step(X200)

Fig. 8. The volume fraction of ˛′ phase as a function of time for c0 = 0.3 (a) andc0 = 0.45 (b) aging at 540 K with an edge dislocation.

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634 Y.-s. Li et al. / Materials Science an

0 = 0.45, applying a strain has practically no effect on the varia-ion of Vf with time: as seen in Fig. 8b, the variation is practicallyhe same with or without the applied strain. This indicates that thelloy phase decomposition has transformed from the nucleation tohe spinodal decomposition as the concentration increases. Thus,he effects of applied strain on the nucleation are greater than onhe spinodal decomposition, and it makes the phase decompositionaster as the alloy concentration decreases.

. Conclusions

The combination effects of applied strain and dislocation onhe precipitation behavior and microstructure evolution of modelinary alloys were investigated using the phase field simulation.he simulations show that an applied tensile strain elongates therecipitates, and that the elongation direction is perpendicular tohe applied strain direction for the soft precipitates, even if the dis-ocations exist. Although the precipitates nucleate at the tensiletress regions of the dislocations and dislocation walls, the orienta-ion and the local morphology of the precipitates at the dislocationegions depend on the relative magnitude of the applied stress andislocation stress. As the alloy concentration increases, the precip-

tates morphology changes from separated elliptical shape to bandhape under the applied strain. Finally, the applied strain makes thehase decomposition faster as the alloy concentration decreases.aken as a whole, these results suggest that one might be able toontrol the microstructure of an alloy and its evolution by suit-ble combination of applied stress, dislocations and composition,nd the theoretical calculation is helpful in predicting what thoseombinations should be.

cknowledgments

We appreciate the helpful discussion with Prof. Tong-Yi Zhangf Hong Kong University of Science and Technology, and Miguel

[[

ineering A 528 (2011) 8628– 8634

Fuentes-Cabrera for insightful discussions. We acknowledge thefinancial support by National Natural Science Foundation of China(No. 51001062), National Basic Research Program of China (No.2011CB605504), and NUST Research Funding (No. 2010ZDJH10).

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