Atomic Mobilities and Diffusivities in the Fcc, L12 and B2 Phases of the Ni–Al System

7/23/2019 Atomic Mobilities and Diffusivities in the Fcc, L12 and B2 Phases of the Ni–Al System

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Lijun Zhanga,b, Yong Dua, Qing Chenc, Ingo Steinbachb, Baiyun Huanga

aState Key Laboratory of Powder Metallurgy, Central South University, Hunan, Chinab Interdisciplinary Centre for Advanced Materials Simulation, Ruhr-Universität Bochum, Bochum, GermanycThermo-Calc Software AB, Stockholm, Sweden

Atomic mobilities and diffusivities in the fcc, L12and B2 phases of the Ni–Al system

A phenomenological model was utilized to describe diffusiv-ities in the c (fcc)/ c’ (L12) and A2/B2 phases of the Ni–Alsystem. An effective strategy, which takes the homogeneityrange and defect concentration into account, was developedin the present work to optimize the atomic mobilities of c’

phase. Such a strategy results in a dramatic decrease in thenumber of atomic mobility parameters to be evaluated forthe L12 phase. The measured composition- and tempera-ture-dependent diffusivities in the Ni–Al system have beenwell replicated by the present mobility descriptions. For theL12 phase, comprehensive comparisons show that with fewermodel parameters the presently obtained mobilities yield abetter fit to experimental diffusivities, compared with pre-

vious assessments. The mobility descriptions are further val-idated by comparing calculated and measured concentrationprofiles for various diffusion couples. The time-dependentAl composition profile for the annealed vapor Al/ c coupleis accurately described for the first time.

Keywords: Ni–Al system; Diffusion; Atomic mobility; Or-der/ disorder phenomena; DICTRA

1. Introduction

Knowledge of both thermodynamic and diffusion charac-

teristics in multicomponent alloys is of critical importancein various materials processes, such as solidification, heattreatment, recrystallization and protective coatings. To gainan insight into the above materials processes, an advanced

computational technique is an appropriate underlying tool.So far, the CALPHAD (CALculation of PHAse Diagram)technique has made significant progress, and a wide varietyof multicomponent thermodynamic databases have alreadybeen constructed [1]. However, this is not the case for diffu-sion data. To simulate diffusion-controlled transformationsin multicomponent systems, the DICTRA (DIffusion Con-trolled TRAnsformations) software package [2] has beendeveloped, and operates under the CALPHAD framework.Based on the sharp interface and local equilibrium hypoth-esis, DICTRA has been successfully utilized to simulatevarious phase transformation processes with the so-calledatomic mobility database [3–7]. Moreover, 2D or 3D simu-lations using the phase-field method including effects of in-terfacial energy, stress, strain and convective transport inliquid, in addition to thermodynamic and kinetic data, re-present a new class of simulation tools in materials science[8–11]. The link to real CALPHAD-type thermodynamicand atomic mobility databases [12–15] is also of invalu-able benefit. In phase-field simulations metastable phasesare also addressed, and thus an extrapolation of thermody-namic functions beyond the region of thermodynamic sta-bility are needed resting on a sound model basis. Therefore,there is an increasing need to establish accurate atomic mo-bility databases for advanced material research.

In the present work, we examine the atomic mobilitiesand diffusivities in the c(fcc), c’(L12), and B2 phases of

the Ni–Al system, which is of great technological impor-tance in many applications, such as high-temperature struc-tural materials, coatings, diffusion barriers etc. In this sys-tem, there exists a large amount of experimental data on

L. Zhang et al.: Atomic mobilities and diffusivities in the fcc, L12 and B2 phases of the Ni–Al system

Int. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 12 1461

2 0 1 0 C a r l H a n s e r V e r l a g , M u n i c h

, G e r m a n y

w w w . i j m r . d e

N o t f o r u s e i n i n t e r n e t o r i n t r a n e t s i t e s

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diffusivities in various phases and composition profiles fordifferent diffusion couples. The former is the main input of our CALPHAD-type assessment work and is critically re-viewed in Section 3; the latter is used to validate the ob-tained mobility data in the final part of Section 4. Severalassessments on the atomic mobilities in the Ni–Al system

can be found in the literature. The earliest one, performedby Engström and Ågren [16], dealt with the disordered cphase. Later, Helander and Ågren [6] introduced a phenom-enological model on the basis of Girifalco’s theory [17] andobtained a successful description for the atomic mobilitiesand diffusivities in the ordered B2 phase. The work due toHelander and Ågren [6] was then updated by Wei et al.[18], who considered more experimental data. However,the results by Wei et al. [18] were found to be unreasonableby Liu and Liang [19]. Most recently, Campbell carefullyassessed [20] and reassessed [21] the mobilities and diffu-sivities for both the B2 and L12 phases in the Ni–Al(–Cr)system by adopting the phenomenological model due to

Helander and Ågren [6]. While these latest modeling resultsagree well with most of experimental data, as many as 11[20] and 12 [21] parameters were employed for the L12

phase in the Ni–Al system. The introduction of so many pa-rameters was not justified considering the relatively narrowcomposition range of the L12 phase. Besides, it is recog-nized that the self-diffusion parameters for the pure ele-ments are the “building blocks” for the development of multicomponent atomic mobility databases. However, onlythe self-diffusion coefficients for pure elements at hightemperatures are usually taken into account in most of theearly assessments, such as the assessments by Jönsson [22]for pure Ni, and Engström and Ågren [16] for pure Al. The

direct extrapolation of these self-diffusion parameters ob-tained by means of the data at high temperatures into thelow temperature region may result in inaccurate predictionsof diffusivities. Such an extrapolation should be avoided invarious quantitatively numerical simulations. Cui et al.[23] performed a literature review on the self-diffusivitydata for pure Al over the wide temperature range and sug-gests that a smaller frequency factor than that assessed byEngström and Ågren [16] is likely to be more realistic. Con-sequently, they [23] reassessed the self-diffusion parameterfor pure Al by using more data than those used by Engströmand Ågren [16]. However, they [23] ignored two groups of literature data, i.e. Fradin and Rowland [24] at high tem-peratures and Burke and Ramachandran [25] at low tem-peratures. Using high purity Al (99.9999 wt.%) coupledwith nuclear magnetic resonance (NMR) and radio-fre-quency field pulse technique, Fradin and Rowland [24]measured the self-diffusion coefficients of pure Al at a rela-tively wide temperature range (513 – 823 K), which show acertain deviation from those earlier data by Lundy and Mur-dock [26] and Stoebe et al. [27], but in a good agreementwith those later data by Beyeler and Adda [28] and Messeret al. [29] and also the only two data at low temperaturesfrom Burke and Ramachandran [25] and Volin and Balluffi[30]. Therefore, there is a need to update the self-diffusionparameters for pure Ni and Al by considering all the experi-mental data over the entire temperature range before reas-

sessing atomic mobilities in the c and c’ phases of the pres-ent Ni–Al system.

In the present work, we intend to (I) update the self-diffu-sion parameters for pure Ni and Al by considering all the

experimental data over the entire temperature range; (II) re-assess the atomic mobilities and diffusivities in the fcc, L12

and B2 phases by taking into account almost all availableexperimental data and paying special attention to the num-ber of parameters used for the description of L12 phase,and compare the present results with the previous assess-

ments; and (III) validate the presently obtained atomic mo-bilities and diffusivities by comparing the calculated andmeasured concentration profiles in a variety of diffusioncouples.

2. Models

The thermodynamic description of the fcc, L12 and B2phases in the Ni–Al system is directly taken from the workby Dupin et al. [31]. Since L12 and B2 phases are orderedstructures of fcc and bcc, respectively, they were repre-sented by a two-sublattice model, where Ni and antisite Alatoms occupy one sublattice, and Al and antisite Ni atomsthe other. For the B2 phase, vacancies were introduced intoboth sublattices.

According to the model suggested by Andersson and Åg-ren [32] and Jönsson [22], which is based on the absolutereaction rate theory [33], the atomic mobility for an elementB, M B, can be expressed as:

M B ¼ exp RT ln M 0B

RT

exp

QB

RT

1 RT

mgC ð1Þ

where R is the gas constant, T the temperature, M 0B a fre-quency factor and QB the activation enthalpy. Both M 0Band QB are in general dependent on composition, tempera-

ture and pressure. mgC is a factor taking into account the fer-romagnetic contribution to the diffusion coefficient, andcan be expressed as [34]

mgC ¼ exp 6nð Þ exp nQB

RT

ð2Þ

in which n represents the state of the magnetic order(0 < n < 1), and can be treated as a constant (0.3 for bccalloys, while 0 for fcc alloys [35]).

For a disordered phase, in the spirit of the CALPHAD ap-proach [36], the composition dependency of RT ln M 0B andQB can be represented with the Redlich–Kister expansion,namely

UB ¼Xi

xiUiB þ

Xi

X j>i

xi x jXmr ¼0

r Ui; jB xi x j r " #

ð3Þ

where UB represents RT ln M 0B or QB. UiB is the value of UB

for pure i and r Ui; jB are binary interaction parameters. Ob-

viously, one can simply combine QB and RT ln M 0B intoone parameter, i. e. UB ¼ QB þ RT ln M 0B when the ferro-magnetic effect on diffusion is negligible.

Once the atomic mobilities in an alloy are known, var-ious diffusivities including trace diffusivity, intrinsic diffu-sivity, and chemical diffusivity can be calculated. Assum-ing the monovacancy mechanism for diffusion andneglecting correlation factors [22, 32], the tracer diffusivity D

B is related to the mobility M B by the Einstein relation:

DB ¼ RTM B ð4Þ


1462 Int. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 12

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, G e r m a n y






In the volume-fixed frame of reference or number-fixedframe of reference, the interdiffusion coefficient ~ Dn

kj, whichrelates the flux of element k with the gradient of component j and reference component n, is given by [32]:

~ Dn

kj ¼ X

i

dik

xk

ð Þ xi M

i

qui

q xi

qui

q xn ð5Þ

where dik is the Kroneker delta ( dik ¼ 1 if i ¼ k , otherwise dik ¼ 0) and li is the chemical potential of element i. Theintrinsic diffusion coefficient, Dn

kj, is defined in the lattice-fixed frame of reference and can be identified as [32]

Dnkj ¼ xk M k

quk

q x j

quk

q xn

ð6Þ

For an ordered phase, the composition dependence of theatomic mobility should include also the effect of chemicalordering. Based on Girifalco’s conclusion [17] that the in-

crease of the activation energy due to chemical ordering de-pends quadratically on the long-range order parameter, He-lander and Ågren [6] proposed a phenomenological modelto describe diffusion in phases with the B2 ordering trans-formation, where the activation energy can be expressed as

QB ¼ QdisB þ Qord

B ð7Þ

where QdisB represents the contribution from the disordered

state and can be expressed by Eq. (3), while QordB denotes

the contribution from chemical ordering. This later quantityis given by an equation of the form:

QordB ¼ X

iXi6¼ j

QordBij y

i y b j xi x jh i ð8Þ

where QordBij is a parameter describing the contribution of the

component B as a result of the chemical ordering of the i– jatoms on the two sublattices and yi is the site fraction of component i on the sublattice,

yi ¼ N i N total

ð9Þ

in which N i is the number of sites on the sublattice thatare occupied by an i atom and N total is the total number of sites on the sublattice. Although the phenomenologicalmodel developed by Helander and Ågren [6] was for an

AB-type alloy (B2 structure), it is proved to be applicablealso for the AB3-type alloy (L12 structure) by Tôkei et al.[37].

3. Review of literature data

There exists a large amount of diffusivity data for fcc, L1 2,and B2 phases in the Ni–Al system. To assign suitableweighting for various data from different sources in thepresent assessment, the following geometric relations areused as an important criterion:(I) the inter-diffusion coefficient is equivalent to the im-

purity diffusion coefficient of Al in Ni when xNi? 1,

and equivalent to the impurity diffusion coefficient of Ni in Al when xNi? 0;

(II) the tracer diffusion coefficient of Al is equivalent tothe self-diffusion coefficient of Al when xNi? 0, and

equivalent to the impurity diffusion coefficient of Alin Ni when xNi? 1; and

(III) the tracer diffusion coefficient of Ni is equivalent tothe self-diffusion coefficient of Ni when xNi? 1, andequivalent to the impurity diffusion coefficient of Niin Al when xNi? 0.

3.1. Fcc phase

As listed in Table 1, various experimental diffusion coeffi-cients associated with the fcc phase exist in the literature,including self-diffusion coefficients of Al [24 – 30] and Ni[38–56] DAl

Al and DNiNi

, impurity diffusion coefficients of

Al in Ni [57–59] and Ni in Al [60, 61] DNiAl and DAl

Ni

, as

well as inter-diffusion coefficients [57, 62–72] ~ DNiAlAl

.

Two general criteria were employed in data selection for as-sessment of atomic mobilities of the fcc phase in the presentwork. One is the above mentioned geometric relations, and

the other is that the measured diffusion coefficients at hightemperatures, as well as those at low temperatures but insingle crystals, are used in the assessment. The diffusivitiesresulting from poly-crystals at low temperatures are not uti-lized since the grain boundary diffusion becomes dominantat low temperatures in poly-crystals. The detailed informa-tion for the selection of experimental data is summarizedin Table 1. As for the impurity diffusion coefficients of Niin fcc Al, Du et al. [73] conducted a critical assessment bymeans of the least-squares method and semi-empirical cor-relations. Thus, their results [73] are directly utilized in thepresent assessment.

3.2. L12 and B2 phasesAccording to Campbell [20, 21], the experimental data forNi diffusion as well as the inter-diffusion in the L12 andB2 phases are substantial, while limited experimental dataare available for Al diffusion due to the lack of appropriateisotope. There are two alternative ways to determine theAl tracer diffusion coefficients in the L12 and B2 phases.One way is to measure the solute tracer diffusion coeffi-cients of Al-substituting elements (e.g. In, Ga and Ge) inNi3Al or NiAl, or measure the tracer diffusion coefficientsin similar systems, i.e. Ge in Ni3Ge and Ga in Ni3Ga. Theother way is to use the Darken–Manning relation to deter-mine the Al tracer diffusion coefficients from the inter-dif-fusion or intrinsic diffusion coefficients. Since critical eval-uation of the relevant experimental data have beenperformed by Campbell [20, 21], there is only a need toconcisely summarize and categorize all of them in the pres-ent work. Besides, some literatures not mentioned byCampbell [20, 21] are also included. Table 2 lists all the ex-perimental data in the L12 and B2 phases, including tracerdiffusion coefficients of Al and Ni (Refs. [74–89] for L12

phase, while Refs. [92–96] for B2 phase), intrinsic diffu-sion coefficients for Al and Ni (Refs. [80, 81] for L12 phase,while Refs. [94, 97] for B2 phase), as well as inter-diffusioncoefficients (Refs. [63, 67–71, 79–82, 90, 91] for L12

phase, while Refs. [63, 70, 71, 90, 94, 97–104] for B2

phase).According to Helander and Ågren [6], it is possible to

fully characterize the diffusion in a binary system usingtwo out of the three usual types of diffusivities, i.e. tracer



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diffusion coefficients of two components or tracer diffusioncoefficients for one component plus inter-diffusion coeffi-cients. Thus, the atomic mobilities of Ni and Al in the L12

and B2 phases of the Ni–Al system are assessed based onthe corresponding tracer diffusion coefficients of Ni andthe inter-diffusion coefficients, while the Al tracer diffusion

coefficients in the L12 and B2 phases are not used becausemost of the reported data are not from direct measurements.The only existing experimental information for Al tracerdiffusion coefficients in L12 phase is from Larikov et al.[74], who used the radioactive isotope 26Al with specificactivity of approximately 1.7 · 106 Bq L– 1. Their results[74], however, show certain deviation from those deter-

mined with other methods [75, 79–84]. Considering theirmutual consistency, the experimentally measured Ni tracerdiffusion coefficients in L12 phase from [85, 86, 88, 89],as well as those from [75, 87] above 1173 K, are employedin the present assessment of mobilities in L12 phase, whilethose from [75, 87] below 1173 K are not because the con-

tribution from grain boundary diffusion cannot be negligi-ble in poly-crystal specimens at low temperatures. Due tothe fact that the inter-diffusion coefficients in the L12 phaseabove 1373 K measured by Watanabe et al. [70, 71] fromvarious diffusion couples are consistent with each other,and also in fair agreement with those by Fujiwara and Hor-ita [80, 81], the experimental values above 1373 K from



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Table 1. Summary of various experimental diffusion coefficients in the fcc phase of the Ni–Al system.

Data T (K) Methoda Ref. Codeb

DAlAl 723–923

603–733513–823673–883358–482403–473515–785

26Al, thick layer technique, SC27

Al, NMR line-narrowing measurementNMR and radio-frequency field pulse technique26Al, sectioning, SCTEM, void shrinkage investigationdiffusion controlled climb modelNMR and modified Slichter–Allion theory

[26]

[27][24][28][30][25][29]

&

&

&

&

&

&

&

DNiNi 1273

1143–15211373–14481423–16731423–1673953–1103973–1373

1202–1576948–1023748–923

1173–14731273–16731253–16731173–14731338–15631163–1477813–1193873–1273

63Ni, SAM, PC63Ni, sectioning and SAM, PC63Ni, activity measurement, PC63Ni and 59Ni, sectioning, PC63Ni and 59Ni, SAM, PC63Ni, surface monitoring technique, SC63Ni, sectioning and SAM, SC and PC63Ni, layer-by-layer sample taking63Ni, SAM, SC and PC63Ni, SAM, SC63Ni, absorption method, SC and PC63Ni, graphs of radioactivity distribution63Ni, sectioning and SAM, SC63Ni, absorption method63Ni, Gruzin method, PC63Ni, absorption method63Ni, miscrosectioning technique, SC63Ni, absorption method, SC

[38][39][40][41][42][43]

[44, 45][46][47][48][49][50][51][52][53][54][55][56]

&

&

&

&

&

&

&

&

^

&

&

&

&

&

&

&

&

&

DNiAl 1373–1573

1073–1243914–1212

pressure-welded DC, spectro photometrymeasurement of weight gain by surface oxidationSIMS, SC

[57][58][59]

&

&

&

DAlNi 623–903

776–924entire range

63Ni, sectioning

electrical resistance measurementthe least-squares method

[60]

[61][73]

~

~

&

~ DNiAlAl 1373–1573

15331273–15731273–15731273–14731073–14731073–14731173–14731173–14731 073, 1 273

pressure-welded DC, lathe sectioningsingle-phase DC, BMMNi3Al/Ni and NiAl/Ni DC, BMMvapor/solid single-phase DC, EPMA, BMMsingle-phase DC, EPMA, BMMsingle-phase DC, AEM and EPMA, BMMNi3Al/Ni DC, AEM and EPMA, BMMNiAl/Ni DC, AEM and EPMA, SFTvarious kinds of DC, AEM and EPMA, SFTsingle-phase DC, EMPA, BMM

[57][62][63][64][65][66]

[67–69][70][71][72]

&

&

&

&

&

&

&

&

&

&

a SC = single crystals; PC = poly-crystals; NMR = nuclear magnetic resonance; TEM = transmission electron microscopy; SAM = surfaceactivity method; SIMS = secondary ion mass spectrometry; DC = diffusion couples; BMM = Boltzmann–Matano method; EPMA = elec-tron probe microanalyzer; AEM = analytical electron microscope; SFT = Sauer&Freise technique.

b Indicates whether the data are used or not used in the atomic mobility assessment: &, used; ~ not directly used because the assessmentwork from Du et al. [73] is used; ^, the experimental data from single crystals are used, while those from poly-crystals are not; &, notused but used to compare with the present assessment.


, G e r m a n y






[70, 71] as well as those from [80, 81] are utilized in thepresent assessment. As for the B2 phase, the experimentalNi tracer diffusion coefficients from Frank et al. [96], aswell as those from Hancock and McDonnell [95] above1273 K, are employed in the present assessment, while

those from Hancock and McDonnell [95] below 1273 Kare not employed because there exists considerable grainboundary diffusion at low temperatures. Due to good agree-ments among them, the inter-diffusion coefficients above1173 K from Watanabe et al. [70, 71] and Paul et al. [93,



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Table 2. Summary of various experimental diffusion coefficients in the L12 and B2 phases of the Ni–Al system.

Phase Data T (K) Methoda Ref. Codeb

L12 DAl 1273–1473

1173–1473974–1206

1005–14221400

1423–15231173–13731172–15321173–1373

radioactive Al isotope114mIn, SSM, PC68Ga in Ni3Ga, 67Ga in Ni3Ga, IBSS69/71Ga and 74/72/70Ge in Ni3Al, SIMS, SCDarken–Manning equationManning equationDarken–Manning equationextrapolation from ternary systems67Ga in Ni3Al, RAM

[74][75][76]

[77, 78][79]

[80, 81][82][83][84]

&

&

&

&

&

&

&

&

&

DNi 1460–18301273–1473

1163–1477965–1625

1004–1422

1173–14731183–1373

63

Ni, RAMradioactive Ni isotope63Ni, absorption method63Ni, RAM & IBSS, PC63Ni, SSM, PS (1223– 1 422 K)64Ni, SIMS, SC (1004–1259 K)63Ni, SSM, PC63Ni, SSM

[85][74][86][87][88]

[75][89]

&&

&

^

&

^

&

DAl, DNi 1 423 – 1523 Darken equation & velocity equation [80,81] &

~ DNiAlAl 1273–1573

1143–13731073–14731173–14731173–1473

1273–14731223–14731323–15231173–1373

Ni/NiAl DC, EMPA, BMMvarious kinds of DC, EPMA, WFNi/Ni3Al DC, AEM and EPMA, BMMNi/NiAl DC, AEM and EPMA, SFTvarious kinds of DC, AEM and EPMA, SFT

single-phase DC, PC, EMPA, BMMsingle-phase DC, PC, EMPA, BMMNi/NiAl DC, EPMA, SFTtwo-phase DC, EPMA, BMM

[63][90]

[67–69][70][71]

[79][91]

[80, 81][82]

&

&

&

^

^

&

&

&

&

B2 DAl 1007–1373

1373–16731273

114mIn, sectioning & RAMNiAl/NiAl–In DC, EPMA, Hall’s methodManning’s equation & intrinsic diffusivities

[92][93][94]

&

&

&

DNi 1273–1623

1050–1630

1273

63Ni, RAM, PC63Ni, SSM for HTM;64Ni, SIMS for LTM; SCManning’s equation & intrinsic diffusivities

[95][96]

[94]

^

&

&

DAl, DNi 1471–17701273

DC, markers distribution and Darken equationintrinsic diffusivities and Darken equation

[97][94]

&

&

~ DNiAlAl 1273–1573

1143–12731223–14231173–14731173–14731072–15721173–13731223–17731473–17731123–14231273–1473

Ni/NiAl DC, EPMA, BMMlayer growth in Al coating/Nipack-aluminized Ni/NiAl, EPMA, BMMNi/NiAl DC, AEM and EPMA, SFTvarious kinds of DC, AEM and EPMA, SFTsingle-phase vapor–solid DC, EPMA, BMMsingle-phase DC, EPMA, WFsingle-phase DC, EPMA, SFTsingle-phase DC, EPMA, SFTpacking Al/Ni, EPMA, WFvarious kinds of DC, EPMA, SFT

[63][98][90][70][71][99]

[100][101][97]

[102, 103][94, 104]

&

&

^

^

^

&

^

&

&

&

&

a SC = single crystals; PC = poly-crystals; IBSS = ion-beam sputter-sectioning; SIMS = secondary ion mass spectrometry; RAM = resi-dual activity method; SMM = serial sectioning method; DC = diffusion couples; BMM = Boltzmann–Matano method; EPMA = elec-tron probe microanalyzer; AEM = analytical electron microscope; SFT = Sauer & Freise technique; WF = Wagner’s function;HTM = high temperature measurement; LTM = low temperature measurement.

b Indicates whether the data are used or not used in the atomic mobility assessment: &, used; ^, partly used; &, not used but used to com-pare with the present assessment.


, G e r m a n y






104], those in Ni-rich region from Shankar and Seigle [90],as well as the inter-diffusion coefficients measured by Kimand Chang [100] except for those around stoichiometriccomposition, are included in the present assessment of atomic mobilities in B2 phase.

4. Results and discussion

The assessment of atomic mobilities for the fcc, L12, andB2 phases in the Ni–Al system is conducted phase by phaseby means of the PARROT module [105] in the DICTRA



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Table 3. Summary of atomic mobility parameters of Al and Ni in the fcc, L1 2 and B2 phases obtained in the present work. The param-eters from the previous assessments [6, 16, 21–23] are also listed for comparison.a

Phases Mobility Parameters Sources

fcc Mobility of Al UAlAl ¼ 123 111:6 97:34 T

UAlNi ¼ 268 381:0 71:04 T

0UAlAlNi ¼ 308 067:5 þ 111:52 T

This work

UAlAl ¼ 142 000 72:12 T

UNiAl ¼ 284 000 59:83 T

0UAl;NiAl ¼ 41300 91:2 T

[16]

UAlAl ¼ 126 719 95:08 T [23]

Mobility of Ni UNiNi ¼ 271 377:6 81:79 T

UAlNi ¼ 144 600:0 64:85 T b

0UAl;NiNi ¼ 29571:8

This work

UNiNi ¼ 287 000 69:8 T

UAlNi ¼ 145 900 64:26 T

0UAl;NiNi ¼ 113 000 þ 65:5 T

[16, 22]

L12 Mobility of Al QordAlAlNi ¼ þ0:0, Qord

AlNiAl ¼ 30350:0 This work

QordAlAlNi ¼ 220214 50:98 TQordAlNiAl ¼ 2780 þ 4:61 T Qord

AlAlNiAl ¼ þ475293, QordAlAlNiNi ¼ þ435 391

[21]c

Mobility of Ni QordNiAlNi ¼ þ0:0, Qord

NiNiAl ¼ 55441:9 This work

QordNiAlNi ¼ 674 704 þ 42:63 T

QordNiNiAl ¼ 101 347 3:12 T

QordNiAlNiAl ¼ 222275, Qord

NiAlNiNi ¼ þ612 177

[21]c

B2 Mobility of Al QordAlAlNi ¼ Qord

AlNiAl ¼ 350911:2

QordAlAlVa ¼ Qord

AlVaAl ¼ þ365062:0

QordAlNiVa ¼ Qord

AlVaNi ¼ þ0:0

This work

QordAlAlNi ¼ QordAlNiAl ¼ 359700Qord

AlAlVa ¼ QordAlVaAl ¼ þ1071200

QordAlNiVa ¼ Qord

AlVaNi ¼ þ0:0

[6]

Mobility of Ni QordNiAlNi ¼ Qord

NiNiAl ¼ 329521:7

QordNiAlVa ¼ Qord

NiVaAl ¼ 335668:5

QordNiNiVa ¼ Qord

NiVaNi ¼ þ0:0

This work

QordNiAlNi ¼ Qord

NiNiAl ¼ 314400

QordNiAlVa ¼ Qord

NiVaAl ¼ þ107000

QordNiNiVa ¼ Qord

NiVaNi ¼ þ0:0

[6]

a In J (mole-atoms) – 1; temperature (T ) in Kelvin.b Indicate that this mobility is directly taken from Du et al. [73].cQord

AlAlNiAl , QordAlAlNiNi, Q

ordNiAlNiAl, Q

ordNiAlNiNi correspond to the interactive parameters Dc 0

QAlAl;Ni:Al, D

c 0QAl

Al;Ni:Ni, Dc 0QNi

Al;Ni:Al, Dc 0QNi

Al;Ni:Ni

in Ref. [21], respectively.


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software based on all the literature data critically reviewedin Section 3.

4.1. Atomic mobility of disordered fcc phase

The assessment of the atomic mobilities for fcc phase

started with the two end-members UAlAl and UNiNi

on the ba-sis of the experimental self-diffusion coefficients of Al andNi, as listed in Table 1. Then the end-member, UAl

Ni, corre-sponding to the impurity diffusivity of Ni in Al was takendirectly from a recent work by Du et al. [73]. The remainingend-member, UNi

Al, was optimized with the experimentaldata on the impurity diffusivity of Al in Ni [57, 59]. Allisonand Samelson’s data [58] were not used in the present as-

sessment because considerable grain boundary diffusionhad been observed in their samples. The interaction param-eters in the mobility description were then assessed byusing the experimental inter-diffusion data in Table 1. Thefinally obtained mobility parameters for the fcc phase arelisted in Table 3.

Figure 1a–d presents the calculated self-diffusion coeffi-cients of Al and Ni, and impurity diffusion coefficients of Al in fcc Ni and Ni in fcc Al along with the correspondingexperimental data [24 – 30, 38– 61], respectively. Theprevious assessment results from Engström and Ågren[16], Jönsson [22] and Cui et al. [23], as well as those fromthe most recent first-principles calculations by Mantinaet al. [106, 107], are also plotted in Fig. 1 for comparison.



BBasic

(a) (b)

(c) (d)Fig. 1. Calculated self-diffusion coefficients of (a) Al, (b) Ni, impurity diffusion coefficients of (c) Al in fcc Ni and (d) Ni in fcc Al, compared withthe corresponding experimental data [24– 30, 38 – 61], the previous assessments from Engström and Ågren [16], Jönsson [22], and Cui et al. [23] aswell as those from the most recent first-principles calculations by Mantina et al. [106, 107].


, G e r m a n y






As shown in Fig. 1a, the calculated Al self-diffusion coeffi-cients due to the present work and Cui et al. [23] can notice-ably reproduce more experimental data than Engström andÅgren [16]. But in order to characterize which assessmentis statistically better between the present work and Cui et al.[23] for Al as well as between the present work and Jönsson

[22] for Ni, the coefficient of determination ( R2) are thenemployed. R2 is a statistical measure of how well the re-gression line approximates the real data points, and an R2

of 1.0 indicates that the regression line perfectly fits thedata. In general, all the experimental data are included inthe calculation of R2 except that those from Kalinovichet al. [50] are not used for Ni because they show noticeabledeviations from all the other data, shown in Fig. 1b. Thecalculated R2 from the present work and Cui et al. [23] forAl are 0.99711 and 0.99030, while those of the presentwork and Jönsson [22] for Ni are 0.99068 and 0.98466.The calculations on R2 indicate that the presently obtainedself-diffusion parameters for Al and Ni are both statistically

better than the previous assessments [22, 23]. It is note-worthy that our calculated self-diffusion coefficients of Alare nearly the same as those from first-principles calcula-tions by Mantina et al. [106]. Moreover, the presently ob-tained Al and Ni self-diffusion coefficients have been suc-cessfully applied in establishing the atomic mobilitydatabase of the Al–Cu [108] and Ni–Pt [109] systems. Thecomparisons between the measured inter-diffusion coeffi-cients [57, 62–72] and the calculated ones from the presentwork and Engström and Ågren [16] are shown in Fig. 2.Both assessments can describe most of the experimentaldata, and are close to each other. In comparison with theprevious assessment [16], more experimental data are con-

sidered in the present work. Furthermore, the present evalu-

ation utilizes one less parameter, compared with the pre-vious work [16].

4.2. Atomic mobilities of ordered L12 and B2 phases

According to Eqs. (7–9), four end-members, QordAlAlNi, Q

ordAlNiAl,

QordNiAlNi and QordNiNiAl, can be used to describe the order contri-bution to the activation energy in the L12 phase. Since the dif-fusion in L12 phase occurs via jumps between two differentsublattices, these contributions are not symmetric, as pointedout by Campbell [20, 21]. Therefore, no similar simplificationin B2 phase, like Qord

AAB ¼ QordABA, can be made for the end-

members in L12 phase. However, the selection of parameterscan be made on the basis of different considerations. From athermodynamic point of view, it is easy to know that yNi y

b Al yAl y

b Ni for Ni3Al since antisite defect concentrations

are very small in the ordered state. This means that the con-tributions for the Al:Ni configuration could be negligibleprovided the corresponding parameters, Qord

AlAlNi and QordNiAlNi,

are not exceptionally large. Furthermore, the homogeneityrange of L12 phase is very narrow [31]. Such a small homo-geneity prevents a large number of parameters to be opti-mized in a meaningful way. Consequently, only Qord

AlNiAl andQord

NiNiAl were chosen for the optimization in a preliminary as-sessment. Firstly, Qord

NiNiAl was determined on the basis of theexperimentally measured Ni tracer diffusion coefficients inL12 phase [85, 86, 88, 89], as well as those above 1173 K[75, 87]. The parameter was then temporarily fixed. Next,Qord

AlNiAl was assessed by employing the critically reviewedinter-diffusion coefficients [70, 71, 80, 81]. Finally, Qord

AlNiAland Qord

NiNiAl were optimized simultaneously. It was found thattwo parameters, Qord

AlNiAl and QordNiNiAl, were enough to repro-

duce all the reliable experimental information. As for theB2 phase, the assessment procedure in the present work isthe same as those of Helander and Ågren [6] and Campbell[20, 21], and thus not described here. In order to make themobilities of A2 phase physically reasonable, the mobilitiesof A2 phase proposed by Helander and Ågren [6] were di-rectly employed in the present work according to Liu andLiang [19]. Table 3 lists all mobility parameters for the L12

and B2 phases.Figure 3a and b shows the calculated tracer diffusion

coefficients of Ni in L12 phase compared with the experi-mental data from Refs. [74, 75, 85–89]. The assessment re-sults from Campbell [21] are also superimposed in the plotsfor comparison. As shown in Fig. 3a and b, the present re-sults yield more or less the same fit to the experimental dataas those of Campbell [21] although only 2 parameters wereemployed in the present work, comparing with 12 parame-ters by Campbell [21], who also considered the Al tracerdiffusion coefficients in her assessment. In addition, themodel-predicted tracer diffusion coefficients of Al in L12

phase are presented in Fig. 3c and d along with the experi-mental data from indirect methods [74–84], which werenot used in the present assessment. This plot can be usedto check the reliability of those Al tracer diffusion coeffi-cients measured with indirect methods [74– 84]. As can beseen in Fig. 3c and d, the predicted Al tracer diffusion coef-ficients based on the presently obtained parameters agree

with those data from indirect measurements [74–84] to acertain degree, indicating that those data can give a good es-timation of the real Al tracer diffusion coefficients. The cal-culated inter-diffusion coefficients of L12 phase at 1373,



BBasic

Fig. 2. Comparisons between the measured inter-diffusion coefficients[57, 62–72] and the calculated ones from the present work and En-gström and Ågren [16].


, G e r m a n y






1 423, 1473 and 1523 K from the present work and Camp-bell [21] are compared with the experimental data [63,67–71, 79–82, 90, 91] in Fig. 4a. It can be seen that moreexperimental data can be reproduced by the present work.The extrapolations from high temperatures into low tem-peratures based on the assessed atomic mobilities are made

in Fig. 4b, where the model-predicted inter-diffusion coef-ficients at 1173 and 1273 K are compared with the corre-sponding experimental data [63, 67–71, 79, 82, 90, 91].As can be seen, the presently calculated inter-diffusioncoefficients agree more or less with the measured ones insingle- and two-phase diffusion couples, but deviate greatly



BBasic

(a) (b)

(c) (d)

Fig. 3. (a) and (b) Calculated tracer diffusion coefficients of Ni in L12 phase compared with the experimental data from [74, 75, 85– 89]; (c) and (d)Model-predicted tracer diffusion coefficients of Al in L12 phase along with the experimental data from indirect methods [74– 84]. The previous as-sessment from Campbell [21] is also appended.

(a) (b)

Fig. 4. Calculated inter-diffusion coefficients of L12 phase at (a) 1373, 1423, 1473 and 1523 K, and (b) 1173 and 1273 K are compared with theexperimental data [63, 67–71, 79–81, 90, 91] and the previous assessment [21].


, G e r m a n y






from those in three-phase diffusion couples. This may bedue to the contribution of grain boundary diffusion to theformation and diffusion process of c’ phase in the c / B2three-phase diffusion couples being comparable to the vol-ume diffusion at temperatures below 1273 K, and this ef-fect becomes bigger as the temperature decreases according

to Fig. 4b.Similarly, the calculated Ni and the model-predicted Al

tracer diffusion coefficients in B2 phase are compared withthe corresponding experimental data [92 – 96] in Fig. 5a– d,respectively. As can be seen in Fig. 5a and b, the presentlycalculated Ni tracer diffusion coefficients show a prefer-ence to the experimental data from Frank et al. [96] over awide temperature range by means of single-crystal alloys,while the previous assessments [6, 21] prefer to those athigh temperatures from Hancock and McDonnell [95] bymeans of poly-crystal alloys. It should be noted that atthe time when Helander and Ågren [6] performed the as-sessment only the experimental data from Hancock and

McDonnell [95] were available. The purpose of Fig. 5cand d is the same as that of Fig. 3c and d, i.e. to check thereliability of those Al tracer diffusion coefficients measuredvia indirect methods [92–94]. Figure 6 shows the calcu-lated inter-diffusion coefficients in the B2 phase along withthe experimental data from [63, 70, 71, 90, 94, 101–104]. Itcan be found that the assessments from the present work

and Campbell [21] yield almost the same fit to the experi-mental data, but better than those from Helander and Ågren[6]. Again, it should be noted that some recently experi-mental data, like those from Paul et al. [94, 104], who tookthe effect of molar volume into account, were not availablewhen Helander and Ågren [6] conducted an assessment.

The comparisons between the experimental intrinsic diffu-sion coefficients [97, 104] and the calculated ones fromthe present work and the previous assessments [6, 21] arepresented in Fig. 7. As can be seen, our results provide thebest fit to the experimental data from Paul et al. [94, 104].

4.3. Simulation of diffusion growth in various diffusioncouples

To further validate the presently obtained atomic mobilitiesof fcc, L12 and B2 phases, comparisons between the calcu-lated and the measured concentration profiles in a series of single- and multi-phase diffusion couples were performed.

Figure 8a–d shows the model-predicted concentration pro-files of Ni-8 at.% Al/Ni (c / c), Al/Ni (gas/ c), Ni-23.1 at.%Al/Ni-6.6 at.% Al (c’ / c’) and Ni-48 at.% Al/Ni-54 at.% Al(B2/B2) diffusion couples along with the correspondingexperimental ones [64, 66, 79, 100], respectively. The de-tailed annealing conditions can be found in each plot. InFig. 8a, we assumed that two equal c regions each with a



BBasic

(a) (b)

(c) (d)Fig. 5. (a) and (b) Calculated tracer diffusion coefficients of Ni in B2 phase compared with the experimental data from [94– 96]; (c) and (d) Model-predicted tracer diffusion coefficients of Al in B2 phase along with the experimental data fromindirect methods [92– 94]. The previous assessmentsfrom Helander and Ågren [6] and Campbell [21] are also appended.


, G e r m a n y






100 lm thickness form the diffusion couple, and a doublegeometric grid was employed to define a higher density of points at the interface. The compositions were defined bythe Ni-8 at.% Al/Ni end members in the mole fraction of

Al using the heavy-side function. For the vapor/solid diffu-sion couple, Al/Ni (gas Al/ c), Yamamoto et al. [64] ob-tained the experimental data based on two assumptions.One is that Al concentration in the surface coincides almost



BBasic

(a)

(b)

(c)

Fig. 6. Calculated inter-diffusion coefficients in the B2 phase (a) along xAl = 0.38; (b) at 1 173, 1 273, 1 373, 1 473, 1 573, 1 673 and 1 773 K;(c) at 1123, 1223, 1323, 1423 and 1623 K, compared with the corre-sponding experimental data [63, 70, 71, 90, 94, 101–104], and the pre-vious assessments [6, 21].

(a)

(b)

(c)Fig. 7. Calculated (a) intrinsic diffusion coefficients rate and intrinsicdiffusion coefficients of (b) Ni and (c) Al in the B2 phase compared withthe experimental data [94, 97, 104] and the previous assessments [6, 21].


, G e r m a n y






with the solubility limit of fcc phase (Ni-rich) in the Ni–Alphase diagram, while the other is that the change in molarvolume of fcc phase shows a linear increase with Al con-centration, i.e. from 6.586 · 10 – 6 m3 mol – 1 for pure Ni to9.995 · 10 – 6 m3 mol– 1 for pure Al. The above two assump-tions were directly adopted in the present simulation to re-

produce their experimental results, as shown in Fig. 8b.The vapor/solid diffusion couple was simulated assumingsingle c region with a total 250 lm thickness, and pure Nias initial composition. A geometric grid was utilized to seta higher density of points at the lower side of the diffusioncouple. In accordance with the assumption from Yamamotoet al. [64] that Al concentration in the surface coincides al-most with the solubility limit of fcc-Ni, we fixed the com-position at the lower side as 19.2 at.% Al as a boundarycondition over the entire simulation process. For the c ’ / c’diffusion couple shown in Fig. 8c, it consists of one c’region with a 80 lm thickness and another one with a85 lm thickness. A double geometric grid was employed

to define a higher density of points at the interface, and thecompositions were defined by Heaviside step functionsusing the equilibrium site fractions corresponding to theNi-23.1 at.% Al/Ni-6.6 at.% Al end members. The samemethod as that used for the c’ / c’ diffusion couple was alsoused to define the simulation condition for the B2/B2 diffu-sion couple in Fig. 8d except for different couple thick-nesses and different compositions. As shown in Fig. 8, allthe model-predicted concentration profiles in various sin-gle-phase diffusion couples agree well with the correspond-ing experimental data.

The comparisons between the calculated concentrationprofiles in multi-phase diffusion couples, i.e. Ni-25 at.%

Al/Ni (c’

/ c) and Ni-45 at.% Al/Ni (B2/ c) annealed at

1173 and 1473 K, and the experimental ones [63, 68, 69]are presented in Fig. 9. The c’ / c diffusion couple in Fig. 9awas assumed to be made up of one c’ region with a 70 lmthickness and another c region with a 90 lm thickness. Eachregion has a geometric grid with a higher density of gridpoints at the interface. The initial compositions for the two

end-members are represented by the equilibrium site frac-tions corresponding to 25 and 0 at.% Al, respectively. Thec’ / c diffusion couple in Fig. 9b was also defined in the simi-lar way as Fig. 9a, but with different region thicknesses. Asfor the B2/ c diffusion couple in Fig. 9c, the simulation as-sumed that one c region with a thickness of 500 lm and an-other B2 region with a thickness of 800 lm form the diffu-sion couple, and each region has a geometric grid with ahigher density of grid points at the interface. The initialend-member compositions are 45 and 0 at.% Al. The compo-sitions are converted into site fractions during the simulation.The c’ phase was assumed to be active, and added with a verythin layer (i.e. 1 lm) at the B2/ c interface. The initial com-

position of c’ phase was assumed to be 25 at.% Al, and de-fined in site fractions. As can be seen in Fig 9, again, all the

model-predicted concentration profiles in various multi-phase diffusion couples agree reasonably with the corre-sponding experimental data. Consequently, the plots in Figs.8 and 9 clearly indicate that the presently obtained atomicmobilities of Al and Ni in fcc, L12 and B2 phases are reliable.

5. Conclusions

. all the related experimental data over the entire tem-perature range, the self-diffusion parameters for pureNi and Al were both updated in the present work. The

comparisons of the coefficient of determination ( R2

) be-



BBasic

Fig. 8. Model-predicted concentration pro-files of (a) solid/solid Ni-8 at.% Al/Ni, (b)vapor/ solid Al/ Ni, (c) solid/solid Ni-23.1 at.% Al/Ni-26.6 at.% Al and (d) solid/ solid Ni-8 at.% Al/Ni-54 at.% Al diffusioncouples along with the corresponding experi-mental ones [64, 66, 79, 100].


, G e r m a n y






tween the present work and the previous assessments[16, 22, 23] clearly indicate that the present assessmentsfor pure Ni and Al are both statistically better;

. On the basis of the phenomenological model proposedby Helander and Ågren [6], atomic mobilities in fcc,L12 and B2 solid phases were reassessed by taking intoaccount all the reliable experimental data via DICTRAsoftware. An effective strategy was successfully uti-lized in the present work to select parameters of or-dered contributions, resulting in a dramatic decreasein parameter usage for the L12 phase. Comprehensivecomparisons between various experimental diffusiv-ities and the presently calculated results, as well as be-tween the previous assessments [6, 16, 21] and thepresent assessment, were also made for fcc, L12 andB2 phases in the present work;

. The present mobility descriptions were validated by com-paring calculated and measured concentration profiles of a variety of diffusion couples. Moreover, the time-depen-dent Al composition profile for the annealed vapor Algas/ c couple is accurately described in the present work.

The financial support from the Chinesisch-Deutsches Zentrum für Wis-senschaftsförderung (Grant No. GZ522), the Creative Research Groupof National Natural Science Foundation of China (Grant No. 50721003)and the Key Program of the National Natural Science Foundation of Chi-na (Grant No. 50831007) is acknowledged. One of the authors (Yong Du)acknowledges Cheung Kong Chair Professorship released by Minister of Education of China for financial support. The authors would also like toexpress their thanks to Dr. Anders Engström from the Thermo-Calc Soft-ware AB in Sweden for helping us with his valuable calculations. Specialthanks are given to Dr. C.E. Campbell from National Institute of Stan-dards and Technology, USA for her kind communication with her mostrecent database, and sharing with the literature on self-diffusion coeffi-cient fore pure Ni.

Appendices

While this manuscript was under review, we were discuss-ing with Dr. C. E. Campbell from NIST, USA about estab-lishment of a reference database for the self-diffusion mobi-lities in pure materials. She kindly provided us with somemissing literature for Ni self-diffusion coefficients [110–116]. In order to check the reliability of the presently ob-tained Ni self-diffusion parameters, we compared those ex-perimental self-diffusion coefficients [110–116] with thepresently calculated ones, as shown in Fig. 10. It was found



BBasic

Fig. 9. Comparisons between the calculatedand measured [63, 68, 69] concentration pro-files in multi-phase diffusion couples: (a) sol-id/solid Ni-25 at.% Al/pure Ni at 1173 K,(b) solid/ solid Ni-25 at.% Al/ pure Ni at1173 K, and (c) Ni-45 at.% Al/pure Ni an-nealed at 1 173 and 1473 K.

Fig. 10. Calculated self-diffusion coefficients of Ni compared with theexperimental data [110116] provided by Dr. C.E. Campbell fromNIST, USA.


, G e r m a n y






that the current parameters can describe those experimentaldata [110–116] very well though they were not used in ourprevious assessment, indicating that the present Ni self-dif-fusion parameters should be reliable.

Moreover, a statistical analysis was also performed be-tween the present work and Jönsson’s assessment [22] by

taking into account all the experimental self-diffusion coef-ficients of pure Ni except those from Kalinovich et al. [50],which show noticeable deviations from the other data [39–50, 52–56, 110–116]. The calculated coefficient of deter-mination, i.e. R2, for the present work and Jönsson’s assess-ment [22] are 0.99019 and 0.98566, respectively. It alsosuggests that the present work is statistically better thanJönsson’s assessment [22].

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(Received September 9, 2009; accepted September 10, 2010)

Bibliography

DOI 10.3139/146.110428Int. J. Mat. Res. (formerly Z. Metallkd.)

101 (2010) 12; page 1461–1475# Carl Hanser Verlag GmbH & Co. KGISSN 1862-5282

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Professor Dr. Yong DuState Key Laboratory of Powder MetallurgyCentral South University, Changsha, Hunan 410083, P. R. ChinaTel.: +86 731 88836213Fax: +86 731 88710855E-mail: [email protected]

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