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Acta Materialia 59 (2011) 3287–3296

On the effect of superimposed external stresses on the nucleationand growth of Ni4Ti3 particles: A parametric phase field study

Wei Guo a, Ingo Steinbach a,⇑, Christoph Somsen b, Gunther Eggeler b

a ICAMS Institute, Ruhr-University, Bochum, Germanyb Institute for Materials, Ruhr-University, Bochum, Germany

Received 29 November 2010; received in revised form 31 January 2011; accepted 1 February 2011

Abstract

The effect of a superimposed stress on the coarsening of interacting Ni4Ti3 particles is studied using the multi-phase field method. It isfound that the thickness/diameter ratio of a Ni4Ti3 particle in a (1 1 1)B2 plane increases with an increasing [1 1 1]B2 stress component.The particle shape can change from a disk to a sphere with increasing applied stress. It is also found that diffusional and mechanicalinteractions between two Ni4Ti3 particles can promote the nucleation of new particles. This provides an explanation for the autocatalyticnature of nucleation reported previously. Compressive stresses along [1 1 1]B2 increase the volume fraction and growth velocity of theNi4Ti3 particles of the (1 1 1)B2 plane. Misoriented particles disappear during particle growth. The simulation results are discussed inthe light of previous experimental results.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Multi-phase field; Shape memory alloy; Ni4Ti3 precipitation; Applied stress; Diffusion

1. Introduction

NiTi shape memory alloys combine good functional andstructural properties. They are therefore frequently appliedin engineering and medical technology [1–3]. There is aninterest in Ni-rich near-equiatomic NiTi shape memoryalloys because phase transition temperatures can be adjustedthrough the Ni content. The shape memory effect in near-equiatomic NiTi alloys results from the reversible martens-itic transformation between the high temperature phase B2and the low temperature martensitic phase B190. During pro-cessing, NiTi alloys are generally subjected to solutionannealing and subsequent aging. Often a metastable stoichi-ometric phase, Ni4Ti3, precipitates coherently from the B2matrix. Microstructures containing precipitates of Ni4Ti3have received interest because they significantly influencethe martensitic transformation. This is due to three impor-tant factors: the stress state around the precipitates, the Ni

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.02.002

⇑ Corresponding author. Tel.: +49 234 32 27211; fax: +49 234 32 14989.E-mail address: ingo.steinbach@rub.de (I. Steinbach).

depletion in the matrix close to the precipitates and the spa-tial arrangement and distribution of the precipitates. Thestress field surrounding Ni4Ti3 precipitates is related to thelattice mismatch between Ni4Ti3 and B2. This stress fieldaffects the formation of B190 and moreover promotes thenucleation and growth of a new martensitic phase referredto as R phase. The martensitic transformation path changesfrom a one-step (B2! B190) to a two-step transformation(B2! R! B190). Secondly, the local martensitic transfor-mation temperature is strongly affected by the variation inNi concentration in the matrix due to precipitation. Finally,when driving forces are low, Ni4Ti3 preferentially nucleatesnear grain boundaries while regions in the grain interiorremain precipitate-free. This heterogeneous microstructurewith heterogeneously distributed Ni4Ti3 can result in multi-ple step martensitic transformations during cooling. To sim-ulate the precipitation of Ni4Ti3 from oversaturated Ni-richB2 properly, the effect of the coherent stress field, Ni deple-tion and the spatial distribution of particles need to be takeninto account. The phase field method, which has emerged asthe method of choice for the evolution of microstructures

rights reserved.

Fig. 1. Illustration of the procedure to determine the driving force forB2! Ni4Ti3 transformation (Dfchem) in the multi-phase field model. cB2 isthe Ni concentration of B2. cNi4Ti3 is the Ni concentration of thestoichiometric phase Ni4Ti3. lNi

B2-Ni4Ti3is the diffusion potential of Ni in B2

and Ni4Ti3.

3288 W. Guo et al. / Acta Materialia 59 (2011) 3287–3296

over the past few decades, represents a promising tool for thiscomplex task. Extensive phase field simulations of thegrowth of precipitates in near-equiatomic NiTi shape mem-ory alloys have been performed [4–8]. However, these studiesaddress Ni4Ti3 precipitates as solid solutions as opposed tostoichiometric compounds. In the multi-phase field modelused in the present work, we use a consistent treatment ofstoichiometric phase [9]. The objective of this work is toimprove the understanding of the nucleation, growth andinteraction of Ni4Ti3 particles in the B2 matrix. The growthof one Ni4Ti3 particle coherently precipitated in a single crys-tal B2 matrix will be simulated. Subsequently, the effect of asuperimposed external stress on its growth will be studied.Finally, the growth and interaction between two Ni4Ti3 par-ticles is considered and it is shown how this can promoteautocatalytic nucleation sequences. We discuss the predic-tions of our model in the light of previous experimentalfindings.

2. Simulation method

2.1. Multi-phase field model

In the present work, we use the multi-phase field method[10], which was designed to simulate multi-phase, multi-component material systems. The underlying model equa-tions for the phase fields, concentrations and elastic strainfields are:

_/a ¼ �X~N

b¼1

Mab

~N

dFd/a

� dFd/b

� �ð1Þ

c*� ¼ r

XN

a¼1

MdiffrdF

d c*

!ð2Þ

0i ¼ rjrij ¼ rj dFdeij

ð3Þ

where /a is the phase field variable that indicates the lo-cal a volume fraction, Mab represents a measure for themobility of the a=b interface, ~N represents the numberof phases, F is the Helmholtz free energy of the system,c is the local concentration, Mdiff is the diffusive mobil-ity, and rij and eij represent the stress and strain tensor,respectively. The free energy F of our system has threecomponents:

F ¼Z

Xðf inf þ f chem þ f elÞdv ð4Þ

f inf ¼XN

a;b¼1

4rab

gab

g2ab

p2jr/a:r/bj þ /a/b

( )ð5Þ

f chem ¼XN

a¼1

/afaðc*

aÞ þ l*

c*�

XN

a¼1

/a c*

a

!ð6Þ

f el ¼ 1

2

XN

a¼1

/aðeija � e�ija ÞCijkl

a ðekla � e�kl

a Þ ð7Þ

where f inf is the local free energy of the a=b interface, f chem

is the chemical free energy, f el is the elastic free energy, rab

is the interfacial energy, gab is the interface width, l*

is theLagrange multiplier and Cijkl

a represents the tensor with theelastic constants of phase a. More details are given in Ref.[11].

2.2. Model of the stoichiometric phase

Ni4Ti3 is a stoichiometric phase with a small stabilityrange and a needle-like f(c) curve (see Fig. 1). As the com-position deviates from the stoichiometric composition, thefree energy increases rapidly. Because the slope of such freeenergy is undefined at the cusp it is not possible to treat thestoichiometric phase by the condition of equal diffusionpotentials as described in Ref. [12]. Instead, we derivequasi-equilibrium composition at the interface betweenthe B2 matrix and the Ni4Ti3 precipitate and the drivingforce for the transformation from the chemical free energyf chem of the system:

f chem ¼ /B2f chemB2 ðcB2Þ þ /Ni4Ti3

f chemNi4Ti3ðcNi4Ti3Þ ð8Þ

where f chema ðcaÞ is the chemical free energy of phase a and ca

is the Ni concentration of phase a. a is B2 or Ni4Ti3. The Niconcentration of the mixture of B2 and Ni4Ti3 is:

c ¼ /B2cB2 þ /Ni4Ti3cNi4Ti3 ð9Þ

where /B2 and /Ni4Ti3are the phase fields of B2 and Ni4Ti3,

respectively. cB2 and cNi4Ti3 represent the Ni concentrationsof B2 and Ni4Ti3. Since c, /B2 and /Ni4Ti3

are known fromthe solutions of Eqs. (1) and (2) for every time step andcNi4Ti3 is constant (57.1 at.%), cB2 can be calculated fromEq. (9).

After finding cB2, the driving force Df chem is obtained bythe construction shown in Fig. 1. First, a tangent line EF isdrawn at the free energy composition curve of B2 at cB2.Then, a line GH across the point corresponding to the freeenergy of Ni4Ti3 can be constructed parallel to EF. Thedriving force for the B2! Ni4Ti3 transformation is then

Fig. 2. Simplified phase diagram used in our simulation. Simulations areperformed for 823 K. The equilibrium concentration of B2 is 50.25 at.%(Ni). Ni4Ti3 is a stoichiometric phase with 57.1 at.% Ni. The slope AB is453.3 K (at.%)�1. The slope CD is 4.5 � 103 K (at.%)�1.

W. Guo et al. / Acta Materialia 59 (2011) 3287–3296 3289

obtained as the distance between the two parallel lines,which can be calculated as:

Df chem ¼ f chemB2 ðcB2Þ � f chem

Ni4Ti3ðcNi4Ti3Þ

þ lNiB2�Ni4Ti3

ðcNi4Ti3 � cB2Þ ð10Þ

where lNiB2-Ni4Ti3

is the diffusion potential equal to the slopesof the lines EF and GH.

Eqs. (8)–(10) precisely define the matrix composition inthe quasi-equilibrium state in the presence of a stoichiom-etric precipitate and the associated relative bulk-free energydifferences. It should be emphasized that in particular whenapplied to stoichiometric phases the free energy has to beevaluated for the phase concentrations and not for the mix-ture concentration, which is continuous between the phaseconcentrations and crosses a region in concentration spacefor which the free energy of the stoichiometric phase isundefined. The present model, which is based on splittingthe concentration field into the phase concentrations,avoids this problem and can correctly evaluate the concen-tration of the stoichiometric phase.

2.3. Chemical energy

The chemical free energy of the B2 phase was approxi-mated using the CALPHAD database [8]. Ni4Ti3 is a meta-stable phase and thermodynamic data are not directlyavailable. In this work, the chemical free energy of Ni4Ti3is therefore constructed as follows. At T = 823 K, the equi-librium concentration of B2 phase is 50.25 (at.%) [13]. Wedraw a tangent at 50.25 at.% to the free energy–concentra-tion curve (f(c) curve) of the B2 phase. Because of the stoi-chiometric nature of Ni4Ti3, the free energy of Ni4Ti3 is thevalue on the tangent line at c(Ni) = 57.1 at.%(�68,120 J mol�1). According to the NiTi phase diagram,the equilibrium concentration of B2 varies very little withtemperature around 800 K. Therefore, it is assumed thatthe equilibrium concentration is 50.25 at.% at T = 803 K,yielding a Gibbs free energy value of �67,268 J mol�1.We obtain a linear temperature dependence of:

gmðNi4Ti3Þ ¼ �33064:0� 42:6T J mol�1 ð11Þ(We note that in solid state thermodynamics the differ-

ence between Helmholtz free energy F and Gibbs freeenergy G, density of Helmholtz free energy f and densityof Gibbs free energy g are negligible.) A linear phase dia-gram is used in this work as shown in Fig. 2. The slopesmAB and mCD of the lines AB and CD are obtained fromthe NiTi phase diagram as 453.3 K (at.%)�1 and4.5 � 103 K (at.%)�1, respectively.

The transformation entropy DS of the B2 to Ni4Ti3transformation is obtained from CALPHAD, using thedatabase [8] and Eq. (11) for the B2 phase and Ni4Ti3,respectively. We find a value of DS = 8.615 � 104 J m–

3 K–1. The chemical driving force is approximated as:

Df chem ¼ ðc� ceqB2ÞmABDS ð12Þ

2.4. Eigen strain

The lattice structures of the B2 phase and Ni4Ti3 are B2and rhombohedral structures respectively (Fig. 3). The lat-tice constant of B2 is aB2 = 0.3014 nm [14]. The lattice con-stants of Ni4Ti3 are aR = 0.6697 nm and aR¼ 113:84� [15].The lattice correspondence between B2 and Ni4Ti3 wasproposed as [16]:

aH==½2 1�3�B2; bH==½�13�2�B2; cH==½111�B2 ð13ÞaH, bH and cH are the unit vectors of the hexagonal latticeof Ni4Ti3. (A hexagonal structure is used here because it iseasier to express the lattice relationship between the B2phase and Ni4Ti3.) The following relations link the hexag-onal and rhombohedral structures [17]:

aH ¼ aR � cR ð14ÞbH ¼ bR � cR ð15ÞcH ¼ cR ð16Þ

aR, bR and cR are the unit vectors of the rhombohedral latticeand are approximately parallel to ½20�1�B2, ½�120�B2

and½0�12�B2. ½14�5�B2, ½�32 1�B2 and ½111�B2 are chosen as thex1, x2 and x3 axes of the simulation coordinate system. Un-der such coordinates, the Eigen strain of the B2 to Ni4Ti3transformation is:

�0:00438 0 0

0 �0:00438 0

0 0 �0:02688

0B@

1CA ð17Þ

Ni4Ti3 has eight variants that can be divided into fourpairs. Each pair is associated with one of the {1 1 1}B2

planes. Because the properties of the variants from the samepair are similar, the eight variants are simply represented byfour variants in the present work. The orientation relation-ships between the four variants are shown in Fig. 4. TheEigen strain of variant 1 is shown in Eq. (17). The Eigenstrains of the other three variants were obtained by coordi-nate rotations.

3. Input parameters and post processing

In the following we consider an alloy with 51 at.% Ni,which is subjected to annealing at 823 K a temperature

Fig. 3. Unit cells of B2 and Ni4Ti3.

Fig. 4. Schematic illustration of four Ni4Ti3 variants V1–V4, related tofour {1 1 1}B2 planes.

Table 1Elastic constants in (GPa) of B2 and Ni4Ti3 used in the simulation [20].

c11 c12 c13 c14 c15 c33 c44

Ni4Ti3 218 143 125 �11 6 255 36NiTi (B2) 175 145 0 0 0 0 35

3290 W. Guo et al. / Acta Materialia 59 (2011) 3287–3296

which is just a little higher than aging temperature used forheat treatments. To allow for a quantitative comparison toexperiments performed at 823 K, we will use a similarityanalysis based on the assumption of diffusion-controlledgrowth (see below). The interface energy rab between B2and Ni4Ti3 is assumed to be 0.05 J m�2, a common valuefor the coherent solid–solid interface. The interface mobil-ity Mab = 10�9 m4 J�1 s�1 is chosen as a compromisebetween calculation time and the condition of full diffusioncontrol according to Karma’s thin interface assymptotics[18]. The concentration field c(x) is calculated in Fick’sapproximation where only diffusion in the B2 phase is con-sidered. The diffusion coefficient is related to the chemicalmobility Mdiff in Eq. (2) by the thermodynamic factor

@2f chem

@c2

_c ¼ /B2DNir2cB2 ð18ÞThe Ni diffusion coefficient of B2 is [19]:

DNi ¼ 1:8� 10�8exp�155� 103 J mol�1

RT

� �m2 s�1 ð19Þ

Since stoichiometric Ni4Ti3 grows into the matrix, Nidiffusion in Ni4Ti3 need not be considered. The elastic con-stants used in the simulation are given in Table 1 in Ref.

[20], the elastic constants of B2 are related to the coordi-nates x1 ¼ ½100�B2, x2 ¼ ½010�B2 and x3 ¼ ½001�B2 whilethe elastic constants of Ni4Ti3 are presented with referenceto the original coordinates x1 ¼ ½4�51�B2, x2 ¼ ½�2�13�B2 andx3 ¼ ½�1�1�1�B2.

The domain size in our calculation is 90 � 90 � 90 gridpoints. For our cuboidal one-particle system, the celldimension is dx = 3 nm. The interface thickness is 5dx. Aperiodic boundary condition is used for the diffusion andphase field calculations. A normal expansion boundarycondition is used for the mechanical calculations. The ini-tial size of the precipitate is r = 20 nm. To study the effectof external stress on the precipitate, six different stress val-ues ranging from �250 to �4000 MPa are applied. Thesame input data were used for the two-particle system.For the system with four particles, the cell dimension wasreduced to dx = 1 nm and the initial size of the particleswas taken as r = 10 nm.

4. Results and discussion

4.1. Stress-free growth of Ni4Ti3

The simulation results of the growth of a Ni4Ti3 particleof variant 1 in a single crystal of the B2 phase are shown inFig. 5a and b. An experimental TEM micrograph of Ni4Ti3particles in a Ni-rich NiTi shape memory alloy are pre-sented in Fig. 5c [21]. In agreement with the experimentalobservation in Fig. 5c, the simulated particle (Fig. 5a andb) shows the typical lenticular shape reflecting the mis-match which is characterized by a tensile stress componentin the B2 matrix perpendicular to the particle plane. Thegrowth kinetics of the simulation and experiment can be

Fig. 5. Calculated particle shape in comparison with experiment TEM micrograph. (a) and (b). Simulation results for 10,800 s aging at 823 K (top and sideview). (c) Experimental TEM image from Ref. [21] where a similar alloy was aged at 773 K for 14,400 s. Four particle variants can be distinguished.Particles which are parallel to the TEM foil are highlighted by arrows.

W. Guo et al. / Acta Materialia 59 (2011) 3287–3296 3291

compared on the basis of a similarity analysis, whichaccounts for the temperature related differences in diffusionbetween experiment and simulation. Simulations were per-formed for a temperature of 823 K, 50 K higher than theexperiment in order to have faster diffusion and a shorterprocess time to be able to mimic it by the numerical simu-lations which are bound to an explicit time steppingscheme. This difference in process temperature, however,can easily be resolved applying a similarity analysis.

We assume diffusion-controlled growth, and consider adiffusion distance rD:

rD ¼ffiffiffiffiffiffiffiffiDsts

p¼

ffiffiffiffiffiffiffiffiffiDete

pð20Þ

where D is the diffusion constant and t is the time. The sub-scripts s and e refer to simulation and experiment, respec-tively. The data of experiment and simulation aresummarized in Table 2. Fig. 5a and b illustrates the twoparameters d and t, which represent the diameter and thethickness of the lenticular particle disk. The precipitateshape is determined by an aspect ratio t/d < 1. The temper-ature exposure in the experiments was 36,000 s. Using Eqs.(19) and (20) we can obtain an estimate for the time neededunder the temperature condition of the simulation to ob-tain the same diffusion distance rD. This yields 8400 s.Using this estimate we can calculate the particle geometryand compare it with the experimental finding in Fig. 5c.There is a remarkable agreement for the scaled thicknessof experiment and simulation; a quite large disagreement,however, for the diameter. This can be understood as onlyfor the growth in the direction of the axis of the particle aone-dimensional approximation, as used by the similarityanalysis, is justified. For the growth in direction of the ra-dius a multidimensional diffusion model has to be applied.

Table 2Comparison of the simulated kinetics with the experiment [26].

Simulation Experiment

Temp. (K) 823 773Time (s) 8400 36,000D (nm) 204 900 ± 340T (nm) 51 68 ± 6V (nm3) 1.67 � 106 4.33 � 107

We also note that the interface thickness (5dx or 15 nm) isbig in comparison to the initial precipitate size of r = 20 nmand it can influence the growth kinetics of the particle inthe initial stage. However, according to the simulationsthe shape of the initial precipitate particle changes from asphere to a lenticular shape very quickly. This means thatthe radius of curvature of the flat face of the particle be-comes much larger than the interface thickness in a shorttime. At the outer rim of the particle, however, the curva-ture might not be resolved sufficiently. This error couldalso contribute to the disagreement of simulation andexperiment in the radial direction.

We now compare our prediction from simulation for823 K with experimental data for Ni concentration arounda particle after 14,400 s annealing at 773 K following a sim-ilar procedure as in the discussion of growth kinetics. Theappropriate reference time is 3500 s.1 Fig. 6a shows a color-coded concentration map of Ni around a lenticular precip-itate, Ni concentration range from 50.8 at.% (dark blue) to51.00 at.% (bright yellow). Fig. 6a contains four Ni isocon-centration lines indicating locations of equal concentration.Fig. 6b shows a calculated concentration profile, which isobtained when we plot the Ni concentration along thedashed vertical central line in Fig. 6a. The calculated dataare in good qualitative agreement with the experiment linescan of Fig. 6c [22]. Simulation and experiment show adepletion of Ni around the Ni4Ti3 precipitation duringgrowth, which is less pronounced in the simulations per-formed for a higher temperature than experiment. Thedepletion difference between simulation and experiment isattributed to the change of the diffusion coefficient forthe temperature difference.

The phase field simulation can also rationalize stress dis-tributions around precipitates. We calculated stress fieldswhich have evolved at 823 K after 1500 s when the particlehas the same aspect ratio (thickness/diameter) as the exper-iment and according to the linear elastic theory the value ofthe strain should be independent of its size. The simulation(Fig. 7a and b) shows a good agreement with experiment

1 For interpretation of color in Figs. 3–5, 8–10 and 12, the reader isreferred to the web version of this article.

Fig. 6. Simulated and experimental results for Ni concentration around a Ni4Ti3 particle. (a) Color-coded Ni concentration map. (b) Ni-concentrationfraction along the dashed vertical central line in Fig. 6a. (c) Experimental line scan represented in Ref. [22] (for details see text).

3292 W. Guo et al. / Acta Materialia 59 (2011) 3287–3296

(Fig. 7c). Fig. 7b shows part of the data obtained along thedashed central vertical line in Fig. 7a. There is an abruptjump from positive (0.005) to negative (�0.025) valueswhen we cross the matrix/particle interface. Fig. 7c showsa color-coded stress distribution map, featuring ezz, whichis the stress component perpendicular to the central planeof the lenticular particle disk. The stress value at point 0is 0.006. The difference from the experimental value(0.0109 ± 0.0038) at point 1 in Fig. 7c [23] is only 0.001.As can be seen from the experimental image our simulationresults are in excellent agreement with the experimentaldata.

4.2. Growth of Ni4Ti3 under an applied stress

4.2.1. One-particle systemTo study the effect of an external stress on the fea-

tures of a Ni4Ti3 precipitate of variant 1, compressivestresses along [1 1 1]B2 are applied. Applied stresses rang-ing from �250 to �4000 MPa were considered. Simula-tion results with increasing applied stresses are shownin Fig. 8a–d. The calculations show that the level ofthe applied stress also affects the kinetics of precipitategrowth. Here we focus on the effect of a [1 1 1]B2 com-pressive stress on the particle shape and compare particleshapes for a constant diameter of the lenticular disk (i.e.

the calculations in Fig. 8 were obtained for differentexposure times). In fact for stresses as high as4000 MPa and assuming that plasticity effects can beneglected our simulations predicts spherical precipitateshape with an aspect ratio of 1 (not shown here). Sucha stress level, of course, is unrealistic in experiments.Fig. 9 shows our simulation results on the effect ofsuperimposed [1 1 1]B2 stresses of �250 and –500 MPaon the evolution of the precipitate volume fraction in aone (variant 1) particle/matrix system.

In passing we note that it is difficult to compare thesephase field predictions to experimental data. It is no prob-lem to apply [1 1 1]B2 stresses in Ni-rich NiTi single crystalswith one family of Ni4Ti3 precipitates. However, it is diffi-cult to apply stress higher than 50 MPa at typical agingtemperatures because intensive plastic deformation wouldoccur and alter all thermodynamic, kinetic and microstruc-ture parameters of the system. At a stress as low as50 MPa, Michutta et al. [24] could not observe an increaseof thickness which our simulations predict for stresseshigher than 250 MPa. It should also be noted that theapparent increase in thickness for a particle in a stressedspecimen (T = 823 K) as compared to a particle in astress-free specimen (T = 773 K) as reported in Ref. [25]cannot be attributed to a stress effect alone. Michuttaet al. [24] have shown that aspect ratio increases withincreasing temperature.

Fig. 7. Comparison between simulated and experimental stress components around a Ni4Ti3 precipitate in z-direction ezz. (a) Color-coded stressdistribution map. (b) Evolution of ezz along central vertical dashed line in Fig. 7a. (c) Experimental stress distributions from Ref. [23]. For details see text.

W. Guo et al. / Acta Materialia 59 (2011) 3287–3296 3293

4.2.2. Four-particle system

We now perform a simulation which shows that spe-cific type of stresses promote certain precipitate variants.For this purpose we start out with four spherical precip-itate nuclei associated with Eigen strains which representtheir variant character, Fig. 10a. The radius of the nucleiis 10 nm. Our simulation is performed for 823 K and acompression of 250 MPa is applied. In Fig. 10 the crys-tallographic orientation of the system is selected suchthat [1 1 1]B2 direction is perpendicular to the surface ofthe figure and points up. After 340 s, the four nuclei takeon the characteristic Ni4Ti3 precipitate shapes, Fig. 10b.Already at this early stage, the applied stress favors theprecipitate in the lower left corner of the system whichrepresents a particle of variant 1 (for definition of precip-itate variants see Fig. 4), Fig. 10b. And as can be seen inFig. 10c, it takes no more than 700 s in our simulationuntil the three unfavorably oriented variants have van-ished and only the favorably oriented variant 1 precipi-tate is present. This result of our phase fieldsimulations explains the experimental finding reportedby Michutta et al. [24], who have used a [1 1 1]B2 com-pressive stress aging to create precipitate/matrix micro-structures with only one variant family of Ni4Ti3precipitates, Fig. 11.

4.3. Autocatalytic nucleation

In the last section of this work we consider specific par-ticle arrangements like those published by Khalil-Allafiet al. [26]. A TEM picture of a row of particles is presentedin Fig. 12a [26]. The particle arrangement in Fig. 12a sug-gests that the precipitates nucleate and grow in a collectivemanner. In order to study this further, we consider the twoparticle/matrix systems shown in Fig. 12b (side view) and c(top view).

In the following we analyse the energetic environment ofthe two precipitate/matrix systems of Fig. 12b.

To analyse the interaction between the two particles, thedriving force for the B2 to Ni4Ti3 phase transformation iscalculated. According to Ref. [27], the driving force isexpressed as:

Dfab ¼ Df chemab þ Df el

ab ð21Þ

where Dfab is the total driving force, Df chemab is the chemical

driving force and Df elab is the elastic driving force. The

expression of Df chemab is shown in Eq. (12). The elastic driv-

ing force is as [27]:

Df elab ¼ rklfðe�kl

a � e�klb Þ �

1

2ð½Cklmn

a ��1 � ½Cklmnb ��1Þrmng ð22Þ

Fig. 8. Dependence of the coherent Ni4Ti3 precipitate shape on applied [1 1 1]B2 stress as predicted by phase field simulation (in the absence of hightemperature plasticity): (a) �250 MPa, (b) �500 MPa, (c) �750 MPa, (d) �1000 MPa.

Fig. 9. Increase of precipitate volume fraction as a function of time at823 K. Superimposed [1 1 1]B2 stresses promote the increase of volumefraction.

3294 W. Guo et al. / Acta Materialia 59 (2011) 3287–3296

where e�ija and e�ijb are the Eigen strains for phases a and brespectively, Cijkl

a and Cijklb are the elastic constants of

phases a and b respectively and rkl is the stress field ob-tained from the simulation.

The results are presented in Fig. 13. Fig. 13a shows atwo-dimensional (2-D) section indicating the distributionof the chemical driving force Df chem in our two-particle

precipitate/matrix system. Chemical driving forces (seeEq. (12)) are low as we approach the particles, due to Nidepletion in the matrix next to the precipitates. Fig. 13bshows another 2-D section which rationalizes how thetwo particles affect the mechanical driving forces (see Eq.(22)) for transformation.

Both curves are displayed in Fig. 13c. Two arrows inFig. 13c mark the locations where we find the highest driv-ing forces. The total driving force as the sum of chemicaldriving force and elastic driving force is shown inFig. 13e. The left part of the curve of the total driving forcedecreases from 36 J cm�3 in the direction away from theprecipitate. The driving force of this part is consumed forthe growth of the particle, which has a higher growth veloc-ity close to the outer tip. At �0.03 lm there is a minimumof the total driving force curve. After this minimum thetotal driving force starts to increase slightly. It can beexpected that there exists a maximum of the total drivingforce in the direction of the scan because the total drivingforce value (�26 J cm�3) at the end of the curve ofFig. 13e is higher than the total driving force value(�23 J cm�3) far from the particles (see the value at theupper right corner or lower left corner of Fig. 13c). How-ever, because of the limit of the size of the calculationbox this maximum of total driving force cannot be seenin Fig. 13e. It is important to notice that such a maximumof the total driving force would mark the easiest place for

Fig. 10. Four precipitate/matrix system where each precipitate belongs to one of the particle variants introduced in Fig. 4. (a) Particles (with specificvariant Eigen strains) nucleate as small spheres at 823 K. (b) The precipitates take on their characteristic lenticular shapes after 340 s. The superimposed[1 1 1]B2 stress favors variant 1 precipitate in the lower left part. [1 1 1]B2 direction is perpendicular to the surface of the figure and points up. (c) After 700 sonly the favorably oriented variant is left.

Fig. 11. Scanning electron micrographs of Ni4Ti3 precipitates in Ni-richNiTi as published by Michutta et al. [24]. (a) All precipitate variants formduring stress-free ageing. (b) A superimposed [1 1 1]B2 compressive stressresults in the presence of only one precipitate family of particle variant 1.

W. Guo et al. / Acta Materialia 59 (2011) 3287–3296 3295

the nucleation of the new particles and therefore mostprobably promotes the evolution of precipitation sequenceslike the one shown in Fig. 12a.

5. Conclusions

In the present work we use a multi-phase field model tosimulate nucleation and growth phenomena which occur in

Fig. 12. Experimental and calculated images of special particle arrangements. (aresult. (c) Top view of the simulation result.

a supersaturated NiTi B2 matrix where Ni4Ti3 particlesform. From the results obtained in the present study, thefollowing conclusions can be drawn:

1. Small model systems of one, two and four Ni4Ti3 parti-cles have been simulated by the multi-phase fieldmethod. Simulation results provide a good descriptionof the precipitation morphology as described in experi-mental work.

2. An external stress applied in the [1 1 1]B2 directionaffects the morphology of lenticular precipitates on(1 1 1)B2 planes: with increasing compressive stress, thethickness/diameter ratio increases. Very high stress caneven result in spherical particle shapes.

3. A [1 1 1]B2 compressive stress promotes Ni4Ti3 precipi-tate on the (1 1 1)B2 plane. The volume fraction of theseparticles and their growth rate both increased. Theseparticles will grow at the expense of all other particleswith different orientation.

4. Two closely spaced Ni4Ti3 precipitates which are cou-pled by diffusive and mechanical interactions can pro-mote the nucleation of a third particle. Thisrationalizes autocatalytic nucleation and precipitationsequences which were described previously.

) Rows of particles as observed in Ref. [26]. (b) Side view of the simulation

Fig. 13. Driving force maps in the two-particle matrix system of Fig. 12b and c. (a) Chemical driving force. (b) Mechanical driving force. (c) Total drivingforce (superposition of chemical and mechanical driving force). (d) Driving forces along the dashed line in Fig. 13c. (e) Total driving force along thedashed line in Fig. 13c.

3296 W. Guo et al. / Acta Materialia 59 (2011) 3287–3296

Acknowledgements

This work was conducted within DFG Sonderfors-chungsbereich 459 project A8. The support from the spon-sors of ICAMS, ThyssenKrupp Steel AG, SalzgitterMannesmann Forschung GmbH, Robert Bosch GmbH,Bayer Materials Science AG and Bayer Technology ServicesGmbH, Benteler AG, the state of North Rhine-Westphaliaand the European Community is gratefully acknowledged.

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