59 fe grain boundary diffusion in nanostructured γ-fe–ni

S. V. Divinski1, F. Hisker1, Y.-S. Kang2, J.-S. Lee2, Chr. Herzig1

1 Institut fuÈr Materialphysik, UniversitaÈt MuÈnster, MuÈnster, Germany2 Department of Metallurgy and Materials Science, Hanyang University, Ansan, Korea

59Fe Grain boundary diffusion innanostructured c-Fe ± NiPart I: Radiotracer experiments and Monte-Carlo simulationin the type-A and B kinetic regimes

For the first time, self-diffusion was systematically investi-gated in well-compacted nanocrystalline (grain size d» 80± 100 nm) c-Fe ± 40 wt.% Ni material in a wide tem-perature range (600 ±1010 K) in all Harrison-type kineticregimes. Samples were prepared by sintering the nanocrys-talline Fe ± Ni powder mixture produced by ball milling ofthe component oxides after reduction in hydrogen atmos-phere. The samples revealed a frequently observed bimodalmicrostructure consisting of nano-scaled grains and micro-meter-scaled agglomerates of the nano-grains. Two differ-ent types of short-circuit paths were found to control thediffusionflux in such material. Owing to the applied sensi-tive radiotracer technique Fe diffusion in both types of in-terface boundaries could be successfully characterized bycombining the evaluation of the experimentally determined59Fe diffusion profiles with a Monte-Carlo simulation ofgrain boundary (GB) diffusion. Part I presents the resultsobtained at elevated temperatures in the type-B and A re-gimes. Due to the sample preparation process the GB mo-tion during the diffusion anneal was proven to be negligible.For the first time, it was shown that there exists an inter-mediate stage between the well-known kinetic regimes Band A if

��Dvtp ' d, where Dv is the bulk diffusivity and t

is the time. The corresponding concentration profiles couldbe linearized in the coordinates of ln c vs. y3=2 (c is the layertracer concentration and y is the penetration depth) and theequation to extract the GB diffusion coefficient from thesedata was derived. The limits of the new AB-type stage wereestablished. It was demonstrated that the processing of thenonconventional experimental GB diffusion profiles in ananocrystalline material can be done properly but is moresophisticated than in a coarse-grained material.

Keywords: Nanostructured material; c-Fe ± Ni alloy;Radiotracer diffusion; Grain boundary diffusion of Fe inFe ± Ni alloy; Monte-Carlo simulation

Korngrenzendiffusion von 59Fe in nanokristallinemc-Fe± Ni

Teil I: Radiotracer-Experimente und Monte-Carlo-Modellierung in den Bereichen der Typ A- undB-Diffusionskinetik

Die Selbstdiffusion wurde erstmals systematisch inkompaktiertem nanokristallinen (Korndurchmesser d~ 80± 100 nm) c-Fe-40 Gew.% Ni Material in einem gro-ûen Temperaturbereich (600 ± 1010 K) in allen kinetischenStadien (nach Harrison) untersucht. Die ProbenpraÈparationerfolgte durch Sintern einer nanokristallinen Fe ± Ni Pul-vermischung, die durch Kugelmahlen der Komponenten-oxide nach Reduktion in WasserstoffatmosphaÈre her-gestellt worden war. Die Proben wiesen eine haÈufigbeobachtete bimodale Mikrostruktur auf, bestehend ausKoÈrnern im Nanometerbereich und mikrometergroûen Ag-glomeraten dieser Nano-KoÈrner. Es wurden zwei Artenvon Kurzschlusswegen festgestellt, uÈber die der Diffusions-fluss in diesem Material erfolgt. Aufgrund der verwen-deten empfindlichen Radiotracertechnik konnte die Fe-Diffusion in beiden GrenzflaÈchenarten erfolgreichcharakterisiert werden durch Kombination der Auswertungder experimentell gemessenen 59Fe Diffusionsprofile miteiner Monte-Carlo-Simulation der Korngrenzendiffusion.Im Teil I werden die bei hoÈheren Temperaturen in den Sta-dien B und A erhaltenen Ergebnisse vorgestellt. Wie ge-zeigt wurde, kann eine Korngrenzenwanderung waÈhrendder DiffusionsgluÈhung aufgrund der ProbenpraÈparations-technik vernachlaÈssigt werden. Es wurde erstmals dieExistenz eines Zwischenstadiums zwischen den bekanntenStadien B und A nachgewiesen, wenn

��Dvtp ' d, wobei Dv

der Volumendiffusionskoeffizient und t die GluÈhzeit sind.Die entsprechenden Konzentrazion-Weg Profile ergabensich als linear in der Auftragung ln c gegen y3=2 (c ist dieTracerkonzentration pro Schicht und y die Eindringtiefe).Es wurde eine Beziehung zur Berechnung des Korngren-zendiffusionskoeffizienten aus diesen Daten hergeleitet.Die Grenzen fuÈr das Auftreten des neuen AB Stadiumswurden definiert. Es wurde gezeigt, dass eine korrekteAuswertung der im nanokristallinen Material gemessenenunkonventionellen Korngrenzendiffusionsprofile moÈglichist. Diese Auswertung ist jedoch komplexer als in einemgrobkristallinen Material.

MK_mk17548 ± 4.6.02/druckhaus koÈthen

S. V. Divinski et al.: 59Fe Grain boundary diffusion in nanostructured c-Fe ± Ni, Part I

256 Ó Carl Hanser Verlag, MuÈnchen Z. Metallkd. 93 (2002) 4

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1 Introduction

Fe ± Ni alloys in a nanocrystalline state attracted much at-tention due to their soft magnetic properties exhibiting lowcoercitivity and high permeability [1]. The synthesis of thenanocrystalline Fe ± Ni powders typically includes ballmilling and hydrogen reduction stages [2]. Subsequentcompacting and sintering of the powders allowed to pro-duce Fe ± Ni alloys which pertained a very small grain size(< 100 nm) after annealings up to 1000 K [3]. A stable na-nostructure offers an opportunity to study the diffusivity ofthe nanocrystalline grain boundaries (GBs) within an ex-tended temperature interval using the conventional high-sensitive radiotracer method and the layer sectioning tech-nique for diffusion profile determination. It is importantthat in such a case the penetration depths can amount up toseveral hundreds of microns. The diffusing tracer atomsmay thus sample thousands of individual GBs supplyingstatistically reliable information on the diffusivity of com-mon GBs in the nano-scaled material.

Note that diffusion in nanostructured materials attractedmuch attention in the recent time [4]. However, up to themoment, the key question, namely whether or not the diffu-sivity of GBs in a nano-scaled material is different from theGB diffusivity in `normal' coarse-grained materials, is stillnot fully answered.

A formal classification scheme of GB diffusion in a na-nostructured material was previously outlined [5]. Thecorresponding systematic experiments with an extendedanalysis of the measured diffusion profiles, however, weremissing so far. It would be highly beneficial, however, tocover the different kinetic regimes of the GB diffusion pe-netration in a single investigation on the same material tohave a fundamental experimental verification of this sophis-ticated theory.

Recently, we have initiated a systematic study of the59Fe and 63Ni radiotracer diffusion in nanocrystalline c-Fe ± Ni over an extended temperature interval using theserial sectioning technique. The measured penetration pro-files extended up to about 150 lm, which is very muchlarger than detected in previous diffusion investigationsin nano-structured samples [6, 7]. The form of these pro-files (and, correspondingly, the kinetic regime of the dif-fusion penetration) can be analyzed with very high preci-sion, since penetration profiles over at least three ordersof magnitude in the decrease of the tracer concentrationwere typically measured. In these experiments the relevantvolume penetration depths,

��Dvtp

(Dv and t are the bulkdiffusion coefficient and time, respectively), change fromabout one micron to smaller than one tenth of nanometer,covering the whole spectrum of possible diffusion re-gimes.

Since the measured penetration profiles revealed a so-phisticated shape, which frequently did not satisfy the for-mal conditions of the given diffusion regime, an extendedMonte-Carlo simulation was undertaken to analyze the dif-fusion process in a nano-material.

In Part I of the present investigation 59Fe diffusion in anano-scaled c-Fe ± Ni alloy was examined at temperaturesfrom 750 to 1000 K in the formal conditions of the Harri-son type-B and type-A kinetic regimes [8]. Since in someparticular cases Fe diffusion proceeded in an intermediateregime between the strict type-B and A conditions, the

general findings of the Monte-Carlo study were used toprocess correctly the relevant unusual experimental pene-tration profiles. The main features of the elaboratedMonte-Carlo scheme of simulation of GB diffusion willbe outlined. The whole data on Fe diffusion in nanoc-Fe ± Ni, including the results of the diffusion measure-ments in the type-C conditions, will be presented inPart II.

2 Experimental measurements

2.1 Sample preparation

For the diffusion experiments nanocrystalline c-Fe ±40 wt.% Ni alloy samples were prepared by a powder me-tallurgical method using a mechano-chemically producedc-Fe ± Ni nano-alloy powder [2]. The mechano-chemicalsynthesis of c-Fe ± Ni powder was performed by a hydro-gen reduction process of ball-milled iron oxide (Fe2O3)and nickel oxide (NiO) powders. A mixture of Fe2O3 andNiO with a composition of Fe-40 wt.% Ni was preparedby blending powders of Fe2O3 (7 lm particle size and99.9 wt.% purity) and NiO (7 lm, 99.9 wt.%). The mixturewas ball milled in a stainless-steel attritor mill at a speedof 300 r. p. m. for 3:6� 104 s. The weight ratio of ballsand powder was 50 : 1. Methyl alcohol was added as amilling agent to prevent particle agglomeration during ballmilling. After ball milling, the alcohol was removed by airdrying at 333 K for 2:26� 104 s. The dried oxide cake ob-tained was broken up and sieved down to a grid size of100 lm. The ball milled powder mixture was then reducedin hydrogen atmosphere at 873 K for 3600 s resulting in anano-agglomerate c-Fe ± Ni powder with the grain size of» 30 nm. X-ray diffraction analysis revealed no reflexeswhich could not be attributed to the face-centred cubicFe ± Ni phase.

The powder was then compacted with a pressure of1250 MPa. The powder compacts were sintered at a heat-ing rate of 0.17 K/s up to 1123 K in hydrogen atmosphere,held for 3600 s at this temperature, and subsequentlycooled down at the same rate. As a result, cylindrical sam-ples of 10 mm in diameter and 2.2 mm in height with nearfull density of above 98 % of the theoretical density wereproduced.

The final grain size of the alloy sample was deter-mined by the X-ray diffraction method to be » 80±100 nm which means that grain coarsening had occurredduring the sintering process. It should be noted here thatcontrol anneals near the highest diffusion annealing tem-perature of about 1000 K for prolonged times did notreveal any notable changes in the grain size when checkedby the conventional X-ray diffractometry. This gives an in-dication that there is no pronounced grain boundary mo-tion during the subsequent tracer diffusion anneals. Thisfeature was confirmed by the shape of the measured pene-tration profiles (see the penetration profiles and argumentsbelow).

2.2 Penetration profile detection

One face of the samples was polished by standard metallo-graphic techniques followed by a pre-anneal in high dy-namic vacuum (10ÿ6 Pa) for surface recovery. The radio-



Z. Metallkd. 93 (2002) 4 257

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tracer 59Fe was dropped in form of a dilute HCl solution onthe polished face of the samples. The samples werewrapped into Ta foil to avoid any undesirable contamina-tion. Diffusion anneals were performed in high dynamicvacuum at temperatures from 750 to 1010 K for severaldays. The temperatures were measured and controlled withNi ± NiCr thermocouples with an accuracy of about �1 K.After the diffusion anneal the samples were reduced in dia-meter (at least 1 to 2 mm) by grinding to remove the effectof lateral surface diffusion. The penetration profiles weredetermined by the precision grinding sectioning technique.The activity of each section was determined in a well-typeintrinsic high purity Ge c-detector as described [9].

The uncertainties of individual points on the penetrationprofiles, stemming from the counting procedure and the er-rors of the depth determination, were estimated to be typi-cally less than 10 %.

2.3 Experimental results

Penetration profiles measured in the present work areshown in Fig. 1 as function of the penetration depth in termsof the Gaussian (y2) (a) or the Suzuoka (y6=5) (b) solution[10]. Note the very high sensitivity of the applied radiotra-

cer method, since the decrease in the tracer activity was ty-pically determined over three to four orders of magnitude,Fig. 1. The experimental parameters (temperature T , timet, and the bulk diffusion coefficients Dv) are summarizedin Table 1. The required bulk diffusion coefficients of Fediffusion in the c-Fe-40 wt.% Ni alloy were taken from theinvestigation of Million et al. [11]: D0 � 8:75 ´ 10 ± 4 m2/sand Q � 301:8 kJ/mol.

Several important parameters have to be taken into ac-count to process these profiles. First of all, this is the param-eter �,

� � s � d2��Dvtp

;�1�

which is lower than 1 in the present experiments, see Ta-ble 1 (d is the GB width and s is the segregation factor).Since Fe diffusion in Fe ± Ni alloys is considered, it is rea-sonable to suppose that s is about unity, s � 1, and we willsimply omit s in the current consideration. Strictly speak-ing, the alloy composition may be slightly different in thebulk and at the grain boundary (resulting in s 6� 1), but suchan effect should be small.

The analysis shows that the ratio K of the grain size d tothe penetration depth by volume diffusion into the grainbulk

��Dvtp

, K � d=��Dvtp

, plays a key role now. Theoreti-cally, it is known that two extreme cases exist in depend-ence on the value of K, if, as it usually is the case at lowertemperatures, the GB diffusivity substantially exceeds thatof the bulk, Dgb4Dv, [8]. The Harrison type-A diffusion re-gime exists if d=

��Dvtp

51 which corresponds to a notableoverlap of the diffusion fluxes from the different GBs. Inthis case an effective diffusivity Deff can only be calculatedfrom the experimental profiles, which should be linear inthe coordinates ln c vs. y2. Here, c is the layer concentrationof the tracer at the penetration depth y. Such conditions areindeed realized in the present investigation at T � 1013 K(open circles, Fig. 1a) when d=

��Dvtp � 0:2 (Table 1). All

other profiles correspond to larger values of K, K � 1 at



258 Z. Metallkd. 93 (2002) 4

Fig. 1. Experimental penetration profiles of 59Fe diffusion in c-Fe-40 wt.% Ni as function of y2 (a), y6=5 (b), and y3=2 (c). k is a scaling factor and yis the penetration depth.

Table 1. Experimental parameters of the present diffusion meas-urements. T and t are the temperature and time of diffusion an-neals, Dv is the bulk diffusioncoefficient, d is the averaged grainsize, and � � d=2

��Dvtp �d is the GB width).

T(K) t(s) Dv�m2sÿ1� d=��Dvtp

�

1013 607 860 2:40� 10ÿ19 0.2 6:6� 10ÿ4

921 613 260 6:68� 10ÿ21 1.25 3:9� 10ÿ3

899 20 520 2:55� 10ÿ21 14 3:4� 10ÿ2

881 930 420 1:12� 10ÿ21 2.2 7:7� 10ÿ3

852 1 464 660 2:74� 10ÿ22 40 1:3� 10ÿ2

751 1 467 800 8:91� 10ÿ25 80 2� 10ÿ1

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lower temperatures, see Table 1, and the penetration pro-files are notably curved in the coordinates ln c vs. y2, seeFig. 1a.

On the other hand, if d=��Dvtp

41 (still��Dvtp

4d), thediffusion conditions correspond to the Harrison type-B re-gime of GB diffusion [8] and the penetration profiles shouldgenerally be composed of two parts. The first of them corre-sponds to the direct volume diffusion from the sample sur-face into the grain bulk. Note that the relevant penetrationdepths are smaller than one micron in our case. The second,main part represents GB diffusion and it should be linear inthe coordinates of ln c vs. y6=5 [10]. These conditions are in-deed realized at T � 852 and 751 K (Fig. 1b). Then, theproduct P � d � Dgb can be determined from the inclinationof the corresponding solid lines in Fig. 1b [10].

However, one can see that the shape of the penetrationprofiles at T � 921 and 881 K deviates remarkably fromthe theoretically predicted shape and these are the caseswhen d=

��Dvtp � 1 (see Table 1). A closer look on these

profiles reveals that they show a negative curvature whenplotted against y6=5 (Fig. 1b) and a positive curvature inthe coordinates ln c vs. y2 (Fig. 1a).

It was suggested [12] that under certain conditions of dif-fusion penetration in a nanostructured material a situationmay emerge where the penetration profiles are linear in thecoordinates ln c vs. y3=2. Having replotted the present pro-files against y3=2 one can see that the linearity of the curvesmeasured at T � 921 and 881 K can be followed up to 3 or-ders of magnitude (Fig. 1c). Unfortunately, the conditionsof such a type of diffusion penetration in nanostructuredmaterial were not defined explicitely [12]. Therefore, wehave performed a Monte-Carlo simulation of combinedGB and bulk diffusion in a nanostructured material to studycarefully the transition between the Harrison type-A andtype-B regimes.

3 Monte-Carlo modelling of grain boundarydiffusion

3.1 The applied model

To derive the model we start from the well-known analyti-cal description of Benoist and Martin [13]. Monte-Carlostudies based on this model were performed, e. g., by Murchand coworkers [14, 15] and by Metsch et al. [16]. They alsohave analyzed the transition between the A and B diffusionregimes. However, the situation emerging at d=

��Dvtp � 1

was not analyzed completely up to now, especially with re-spect to the shape of the corresponding penetration profiles.In the present work the attention will be primarily focusedon the shape of the concentration profiles at different valuesof K � d=

��Dvtp

.The atomistic model under consideration is schemati-

cally presented in Fig. 2. Two main jump frequencies willbe used in the present work, see Fig. 2, namely �b and �v,which describe the jumps within the uniform GB slab andthe bulk, respectively. Since we study self-diffusion, thesegregation effects can be omitted from the considerationand therefore the jump frequencies from the GB to the bulk,�bv, and vice versa, �vb, are equal and �vb � �bv � �v.

To simplify the geometry a cubic lattice for both bulk andGB was chosen. The distance between the nearest-neigh-boring sites, a, was chosen to be a � 1 nm. By constructionthis value corresponds to the GB width d � a, since in ourscheme (Fig. 2) only one plane of points represents the GBplane. Due to the trivial relation between the diffusion coef-ficients and the jump frequencies, Dgb � a2 � �b (the corre-lation effects are absent, since random movement of a parti-cle on a grid is considered without specification of theparticular diffusion mechanism), the given value of a usedin the present simulation plays a role of a scaling factor,which allows to recalculate the model jump frequencies �i

and, in principle, can be chosen arbitrary.Due to symmetry arguments the simulation was per-

formed on the two-dimentional (2D) grid. Let the coordi-nate axis y be perpendicular to the external surface and theaxis x lie in this surface. Then, the periodic boundary condi-tions were chosen along the axis x (at � xmax and ÿ xmax)and an infinite sink was assumed at the y � ymax. Usually,ymax was chosen large enough to prevent the situation thatthe diffusing particle will reach this boundary during thesimulation time. Thus, its value does not affect the derivedresults. The plane x � 0 presents the GB plane and thevalue of 2xmax corresponds to the grain size d in the presentsimulation.

The direct application of the random walk of a particle onthe lattice shown in Fig. 2 faces difficulties if Dgb 4Dv (or,equivalently, �b 4�v) due to very long computation timesfor reaching acceptable statistics. This is why in previousMonte-Carlo simulations the ratio Dgb=Dv � 100 wastypically used. However, commonly Dgb4Dv (say,Dgb=Dv � 105 or more). (The situation becomes evenworse if segregation of the tracer atoms is included intoconsideration). In order to simulate GB diffusion with sub-stantial out-diffusion into the bulk (e. g., to study the effectof the grain size on diffusion) it is required to follow manyparticle jumps within the grain volume (say at least106 jumps per particle). However, before entering into thegrain interior the particle will also make about �b=�v jumps



Z. Metallkd. 93 (2002) 4 259

Fig. 2. Schematic model for Monte-Carlo simulation of GB diffusion.The jump frequencies are ms on the surface, mb in the boundary, and mvin the bulk. The jump frequency from bulk to surface and the reverseone are denoted by mvs and ms, respectively. mvb and mbv means the samefor jumps between volume and GB. The circles present the points onthe surface (black), in the GB (grey), and in the bulk (open). � is theGB width in the simulation.

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within the GB slab. These jumps also should be followed.Taking into account appropriate statistics one finds the ne-cessity to simulate in principle 106 � 105 � 107 � 1018 ele-mentary jumps.

To overcome this difficulty the model of Benoist andMartin [13] is extended here. Let us suppose that a particleis within the GB slab and Dgb 4Dv. In the two-dimensionalcase presented in Fig. 2 the a priory probability that theparticle when performing a jump will leave the GB(will jump to the neighboring grain plane) is p0 ��v=��b � �v� � �v=�b51. Then the probability that the par-ticle will still stay within the GB will be P1 � 1ÿ p0. Let uscall such a jump an `̀ unsuccessfulº jump. Then, the prob-ability that the particle will still stay within the GB afterthe sequence of n jumps (thus n consecutive unsuccessfuljumps) is Pn � �1ÿ p0�n. Then, the following proceduremay be suggested to estimate the number N of unsuccessfuljumps of a particle within the GB. Let R be a random num-ber between 0 and 1, R 2 �0; 1�. We propose to estimate Nby the condition that PN becomes smaller than R. Thus Ncan be computed as the `best' integer satisfying the relation�1ÿ p0�N � R,

N � ln R

ln �1ÿ p0��

; �2�

where ln R= ln�1ÿ p0�� denotes the nearest integer to thevalue of ln R= ln�1ÿ p0�. The direct simulation shows thatthe statistical distribution of the number of the unsuccessfulparticle jumps within the GB calculated according toEq. (2) is exactly the same as using simply the one-by-onejump scheme.

Since the particle performs a Brownian movement withwell-defined properties during its jumps within the GB, wecan simply estimate the ad hoc displacement of the particlefrom its initial position by generating a second randomnumber RGauss which should have a Gaussian-type distribu-tion with the known dispersion � � ��

DgbtN

p. Here tN is

the time spent by the particle to perform N jumps withinthe GB with the diffusivity Dgb.

To speed up the calculations of the particle jumps withinthe bulk one can calculate the map of the probabilities p�n�ijfor a particle staying initially at �0; 0� to attain the site�i; j� exactly after n jumps, ij j � jj j � n. Here �i; j� are the2D coordinates of the particle on the 2D grid. Then one onlyneeds to generate one random number to simulate n particlejumps without loss in the diffusion statistics. Such mapswere calculated up to n � 10 and were used in the follow-ing simulation.

In summary, the general procedure to simulate 2D diffu-sion in the system shown in Fig. 2 was as follows: A particlewas situated at first at the upper surface of the simulationblock, y � 0, randomly assigning the coordinate x,ÿxmax � x � �xmax (all numbers are given in a units). Ifthe particle is within the GB slab, x � 0, then the ad hoc sto-chastic number N of jumps within the GB was calculatedaccording to Eq. (2) and the corresponding displacementalong the GB, �y, was computed according to the abovescheme. Then the coordinates were updated as x � �1 andy � y��y (+1 or ÿ1 was chosen by calling the randomnumber generator) and the particle occurred to be withinthe grain interior. The time variable was updated accordingto the time-residence algorithm. One can see that the ap-plied scheme allows to replace the �b=�v calls of the randomnumber generator by only a few corresponding calls to esti-mate the number of unsuccessful jumps N and the resultingdisplacement �y.

The random walk of the particle within the grain interiorwas simulated according to the above-mentioned probabil-ity maps if the distance l from the external surface or fromthe GB was l � 10a and an appropriate generalization ofthe approach of Eq. (2) to the 2D case was used at largervalues of l, l > 10a.

The initial conditions corresponded to the instantaneousdiffusion source. Typically, the diffusion profiles were aver-aged over 107 ± 108 separate particle walks.



260 Z. Metallkd. 93 (2002) 4

Fig. 3. Calculated penetration profiles at Dv � 10ÿ14 m2sÿ1; t � 100 s; Dgb � 10ÿ9 m2sÿ1; � � 10ÿ8 m; and b � 500 against y2 (a), y3=2 (b), and

y6=5 (c) as function of the grain size d: d � 10ÿ7 m (4), 10ÿ6 m (&), and 10ÿ5 m (*).

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The approach, which was derived in the present investi-gation, allowed us to simulate deep diffusion profiles witha decrease in the concentration well over three decades,see Fig. 3, even by using a PC. The availability of deep pro-files is an imperative condition for studying fine features inthe profile shape at the transition from the type-A to thetype-B kinetics.

3.2 Effect of grain size on diffusion in nano-materials

The present simulations were performed for the values ofthe grain size d within the interval 0:1

��Dvtp � d �

10��Dvtp

to cover the transition from the A to B regime.The values of b � d � Dgb=2Dv

��Dvtp

were chosen to be 0.5,1, 5, 500, 5000, and 50 000 by varying Dgb and keepingother parameters constant: Dv � 10ÿ14 m2sÿ1, t � 100 s,and d � a � 10 nm. As an example, the key results calcu-lated at b � 500 are shown in Fig. 3. Three types of profilesare presented: with d � 10ÿ7 m (triangles), 10ÿ6 m(squares), and 10ÿ5 m (circles) (correspondingly:d � 0:1

��Dvtp

,��Dvtp

, and 10��Dvtp

). These calculated pro-files are plotted against y2 (Fig. 3a), y3=2 (Fig. 3b), and y6=5

(Fig. 3c).In the following the two extreme cases will be discussed

first.

3.2.1 Type-A regime conditions

It is well-known that the A regime of the GB diffusion pene-tration corresponds to the pronounced overlap of the diffu-sion fluxes from different GBs into the adjacent bulk, i.e.,if

��Dvtp

4d [8]. The Monte-Carlo results calculated at��Dvtp � 10d (triangles in Fig. 3a) give linear dependenciesin the coordinates ln c vs. y2 which corresponds to the per-fect Gaussian solution of the diffusion equation. The incli-nation of this line defines the effective diffusivity Deff :

Deff � 1

4t ÿ q ln cqy2

� � : �3�

According to the Hart equation [17] this value of Deff is re-lated to the GB and bulk diffusivities:

DHeff � f � Dgb � �1ÿ f �Dv �4�

where f is the volume fraction of GBs in the material,f � d=d. The dependence of the ratio of the calculated val-ue of Deff and that one expected from Eq. (4), DH

eff , is givenin Fig. 4, circles. With increasing d and

��Dvtp � const the

penetration profiles are well linear against y2 up to the val-ues of d=

��Dvtp � 0:3 and Dcalc

eff � DHeff . At larger d the pro-

files become curved in the coordinates ln c vs. y2 and onlya crude estimation of Deff is possible. Note that taking intoaccount only the few first points at the very beginning ofthe penetration profile calculated at d � 7� 10ÿ7 m the es-timation of Deff would be somewhat better. However, in realexperimental conditions some external factors can substan-tially affect the penetration profiles in the near-surface re-gions and we have fitted the entire profiles to re-calculateDcalc

eff .

3.2.2 Type-B diffusion regime

If the grain size increases appreciably and the out-diffusionfluxes from different GBs do not overlap anymore, the pe-netration profiles change qualitatively and show the well-known two-stage form, as it can be seen from the profilesat d=

��Dvtp � 10, circles in Fig. 3c. The first part is linear

in the coordinates ln c vs. y2 and gives us the value of thebulk diffusion coefficient Dv. The most important secondpart is linear in the coordinates ln c vs. y6=5(Fig. 3c) andfrom the inclination of the corresponding line one can cal-culate the GB diffusivity P [10]:

P � d � Dgb � gDr

v

tq

1

ÿ q ln c

qy6=5

� �w : �5�

Generally, the segregation factor s also should appear in thevalue of P, but in the case of self-diffusion considered heres � 1. The specific values of the parameters r, q, w, and gdepend on the value of the parameter b, e. g., r � 0:5,q � 0:5, w � 5=3, and g � 1:308 at b > 104 [18]. Atsmaller values of b they are appropriately changed [18].Although the values of these parameters were derived onlyat relatively large values of b, b > 10 [18], we will usethe relevant formula for 10 < b < 100 [18] to calculate Peven at smaller b.

Figure 4 (squares) presents the ratio Pcalc=dDgb as a func-tion of the relative grain size (Pcalc is the value of P calcu-lated from the corresponding parts of the profiles accordingto Eq. (5)).

The analysis demonstrates that the B regime conditionsare nearly perfectly fulfilled already at d=

��Dvtp � 7. As d

decreases, the error in the P determination increases andPcalc overestimates the value of P by a factor of 2 atd=

��Dvtp � 3. Another fact, which is important from the

experimental point of view, is that the known relevantequations for the calculation of P at the conditions10 < b < 100 (at large d 4

��Dvtp

) can already be usedat b � 2. This incorporates an error less than �15 %which can be accepted in particular GB diffusion experi-ments.



Z. Metallkd. 93 (2002) 4 261

Fig. 4. Grain size dependencies of the ratios of the values calculatedby the Monte-Carlo method and the `true' values of the GB diffusionparameters computed in the different diffusion regimes.

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3.2.3 Type-AB transition regime

The analysis has shown that the penetration profiles becomepronouncedly curved as the grain size increases aboved=

��Dvtp

> 0:3 when plotted in the coordinates ln c vs. y2,revealing a positive curvature, like in Fig. 3a (circles andsquares). As long as the value of d=

��Dvtp

does not exceed4, such profiles show simultaneously an unusual negativecurvature in the coordinates ln c vs. y6=5, see Fig. 3c. Withsuch a profile form these diffusion conditions definitely donot satisfy the common B kinetic regime. Thus, we havean intermediate kinetic regime which we will call here theAB kinetic regime.

The conditions of the AB regime (b > 2, and0:2 < d=

��Dvtp

< 4) may easily occur in real experimentalconditions (see Table 1) and this explains the unusual pro-file shape at T � 921 and 881 K in the present 59Fe tracerexperiments. A numerical procedure is necessary to extractcorrectly the GB diffusivity from such profiles instead ofprocessing these profiles as measured simply in the A or Bregimes, circles or squares in Fig. 4. Physically, this situa-tion corresponds to a moderate overlapping of the bulk dif-fusion fluxes from different GBs. Following the finding in[12] the profiles were replotted against y3=2(Fig. 3b). It isseen that the profile with d=

��Dgbt

p � 1 is almost perfectlylinear in the coordinates ln c vs. y3=2 (Fig. 3b, squares).The present Monte-Carlo study allows to understand thephysical nature of the unusual shapes of the penetrationprofiles measured in the present work and to establish themathematical conditions of the AB regime of the GB diffu-sion penetration.

The inclination of the profile in the coordinates ln c vs.y3=2 determines the GB diffusion coefficient Dgb. Followingthe general treatment of [12] and specifying the exact value

of the numerical factors, we found that

Dgb � 16:65

1� d=d

D0:1v

d0:2 t0:9

1

ÿ q ln c

qy3=2

� �4=3: �6�

The grain boundary width d enters explicitely in Eq. (6) andits value has to be estimated in particular experimental ap-plications. However, the final results only slightly dependon the specific value of d due to the power of 0.2.

The ratio of Dcalcgb calculated according to Eq. (6) from the

model profiles to the `true' value of Dgb is also shown inFig. 4, triangles. As it is seen from Fig. 4, Eq. (6) can wellbe applied in the interval of the ratio K � d=

��Dvtp

around1: 0:7 � K � 3. Moreover, the present analysis explainsthe unusual form of the penetration profiles in Fig. 1 as cor-responding to the intermediate regime AB of the GB diffu-sion penetration and indicates how to calculate Dgb cor-rectly.

To prove this analysis an additional diffusion experi-ment was performed at T � 899 K (i. e., in the temperatureinterval where the AB regime conditions were previouslyreached) but applying a very short annealing time to satis-fy strictly the B regime conditions, see Table 1. As a re-sults, an almost ideal B-type profile was measuredT � 899 K. This profile is plotted against y6=5 (circles, bot-tom axis) and against y3=2 (squares, upper axis) in Fig. 5. Acontinuous curvature of the profile is obvious when plottedvs. y3=2, whereas good linearity of the GB-related part isobserved over almost two decades when plotted vs. y6=5.Note that the profiles in Fig. 5 have been shifted parallelto the y axis with respect to each other to simplify the pre-sentation.

4 Discussion and conclusions

In the present investigation, the Monte-Carlo simulationwas carried out to derive the mathematical background forprocessing the experimental profiles when changing con-tinuously the diffusion conditions from the strict type-B tothe strict type-A kinetics.

Having analyzed the form of the penetration profiles asfunction of the grain size d, the experimental data can nowbe processed according to the given value of K � d=

��Dvtp

.At T � 1013 K the diffusion conditions fully correspondto the Harrison type-A regime with K � 0:2. Thus, the valueof Deff can be extracted from the corresponding profile inFig. 1a. Deff turns out to be much larger than Dv at this tem-perature, see Tables 1 and 2. Thus, with good accuracy onecan rewrite Eq. (4) as

Deff � f � Dgb � d

dDgb: �7�

Assuming that d � 0:5 nm and d � 100 nm one can esti-mate Dgb at this temperature as Dgb � Deff=f . The result isgiven in Table 2.

At T � 921 and 881 K the experiments were performedin the AB transition regime with K ' 1, see Table 1. Apply-ing Eq. (6) to the corresponding penetration profiles(Fig. 1c) and assuming again that d � 0:5 nm the GB diffu-sivities were computed. The results are listed in Table 2.



262 Z. Metallkd. 93 (2002) 4

Fig. 5. Penetration profile of 59Fe GB diffusion at T � 899K as func-tion of y6=5 (circles, bottom axis) or y3=2 (squares, upper axis). y is thepenetration depth. The lines present linear fits in the corresponding co-ordinates.

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At the lowest temperatures in the present investigation,T � 852 and 751 K, and at T � 899 K with short annealingtime, the conditions of the Harrison type-B regime were ful-filled, namely K > 4, large b, and � � 0:1. Therefore, thecorresponding profiles were processed according to Eq. (5)and the GB diffusivities P were calculated, see Table 2. Tohave a comparison with the above computed Dgb data inthe other GB diffusion regimes we have recalculated the re-levant GB diffusion coefficient Dgb as Dgb � P=d, assumingthat d � 0:5 nm. Complete results are summarized in Ta-ble 2.

Analysing the GB diffusion experiments in nanostruc-tured material, the influence of grain boundary motion onthe profile shape has generally to be taken into account.This is important, because otherwise GB motion duringthe diffusion anneal may apparently result in a significantunderestimation of the GB diffusivity [5]. We note that thepresent penetration profiles do not indicate specific featureswhich have to be attributed to the GB motion. According tothe theoretical arguments and the experimental proof [19]GB motion forces the B-type penetration profiles to be line-ar in the coordinates ln c vs. y. This is not observed in the

present experiments. In order to highlight this feature,the profiles measured at T � 852 and 751 K are re-plotted in Fig. 6 against y (open symbols, upper axis)and y6=5 (solid symbols, bottom axis). Instead of anegative curvature in the coordinates ln c vs. y6=5,which is predicted for the profiles affected by the GBmotion [19], a linear dependence of ln c vs. y6=5 isseen and a distinct positive curvature in the coordi-nates ln c vs. y is notable. These features can beclearlydistinguished due to the high quality of the experi-mental profiles with a measured decrease in the tracerconcentration over four decades. Nevertheless, wehave estimated a hypothetical value of the averageGB velocity V by processing the penetration profiles

according to [19]. As a result we obtained that, e. g.,Vt � 65 nm at T � 852 K. Such a value means that thegrain size d should have been increased by a factor of twoduring the diffusion anneal. This, however, was not ob-served by the X-ray diffraction data taken from the samplesafter the diffusion anneal even at T � 1013 K. Therefore,we exclude significant effects from the GB motion on theGB diffusion characteristics deduced in the present investi-gation. The absence of notable grain growth can be ex-plained by the high sintering temperature (1123 K) duringthe sample preparation, which already allowed grain growthfrom �30 to �80 ± 100 nm in grain size.

Having determined Dgb, one can note that according tothe classification scheme suggested in [18] the A0 and B0diffusion regimes are indeed realized in the present experi-ments, since the corresponding GB penetration lengths,Lgb �

��d � Dgb

p= 4Dv=t� �1=4 [18], exceed notably the aver-

age grain size in the nano-material. However, this refiningof the definition of the diffusion regimes does not changethe equations for the penetration profile processing in thepresent case, but introduces only a geometrical factor [5]which for simplicity was taken here as unity.

The complete temperature dependence of the Fe diffusiv-ity in the nanocrystalline c-Fe-40 wt.% Ni alloy will beanalyzed in Part II of the present investigation includingthe results of the measurements in the type-C regime at low-er temperatures.

As we have already mentioned, the samples reveal about2 % residual porosity. This porosity turned out to affect no-tably the type-C penetration profiles, resulting in an addi-tional short-circuit diffusion path, and this is why they willbe analyzed separately, see Part II. The Fe diffusion profilesmeasured in the type-B, type-AB, and type-A kinetic re-gimes reveal no remarkable indications of a second short-circuit diffusion mode. The reasons of such a behavior willbe explained in Part II.

Summarizing the present consideration, we have demon-strated that the processing of the GB penetration profiles cru-cially depends on the diffusion parameters. A sequence of thepower laws with n � 6=3, 6=4, and 6=5 can be observed dur-ing the gradual transition from the Harrison type-A to type-Bregimes on the dependencies ln c vs. yn. Note that in order todistinguish between the stages AB and B (the powers 6=4 and6=5, respectively) from the shape of the penetration profilesonly, these have to be sufficiently long, as it turned out to bethe case in the present radiotracer measurements. In any caseone has to pay attention to the value of the factorK � d=

��Dvtp

to process the penetration profiles correctly.Note that both 6/5 and 6/4 (or 3/2) powers were derived by



Z. Metallkd. 93 (2002) 4 263

Table 2. The results of the profile processing according to the schemesuggested in the present work. The value of the GB width d was takenas 0.5 nm to calculate Dgb in the Harrison type-A and type-B kinetic re-gimes.

T(K) Regime Deff�m2sÿ1� P�m3sÿ1� Dgb�m2sÿ1� b

1013 A 3:59� 10ÿ15 7:2� 10ÿ13 1:5� 103

921 AB 7:1� 10ÿ14 4:2� 104

881 5:4� 10ÿ14 3:8� 105

899 B 8:95� 10ÿ23 1:8� 10ÿ13 2:4� 106

852 5:60� 10ÿ24 1:1� 10ÿ14 5:6� 105

751 2:22� 10ÿ25 4:3� 10ÿ16 1:0� 108

Fig. 6. Penetration profiles of 59Fe GB diffusion in nanocrystalline c-Fe-40 wt.% Ni as function of y6=5 (circles, bottom axis) or y (squares,upper axis). The dashed lines are drawn to indicate the systematic cur-vature of ln c vs. y plots.

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numerical fits of the calculated concentration profiles andthey present only approximate but fruitful solutions in thegiven conditions of the GB diffusion penetration.

We emphasize that the intermediate stage AB can easilybe realized in diffusion experiments on a nanocrystallinematerial at elevated temperatures, in contrast to conven-tional coarse-grained materials. Therefore, one shouldreally take care about the given diffusion kinetics to processthe experimental profiles properly, especially if they are re-latively short and the profile shape cannot be analyzed withthe proper accuracy.

This joint German-Korean project was initiated and supported by theAlexander von Humboldt Foundation, Bonn, Germany. The authors(J. S. L. and Y. S. K.) gratefully acknowledge also the financial supportfrom the Korean Ministry of Science and Technology through the`̀ 2001 National Research Laboratory Programº. The authors are grate-ful to Y. Mishin for reading the manuscript and making valuable com-ments.

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1 Cheung, C.; Djuanda, F.; Erb, U.; Palumbo, G.: Nanostr. Mater. 5(1995) 513.

2 Lee, J.S.; Kim, T.H.; Yu, J.H.; Chung, S.W.: Nanostr. Mater. 9(1997) 153.

3 Knorr, P.; Nam, J.G.; Lee, J.S.: Metall. Mater. Trans. A 31 (2000)503.

4 Gleiter, H.: phys. stat. sol. b 172 (1992) 41.5 Mishin, Y.; Herzig, Chr.: Nanostr. Mater. 6 (1995) 859.

6 Hofler, H.J.; Averback, R.S.; Hahn, H.; Gleiter, H.: J. Appl. Phys.74 (1993) 3832.

7 Herth, S.; Michel, T.; Tanimoto, H.; Eggersmann, M.; Dittmar, R.;Schaefer, H.-E.; Frank, W.; WuÈrschum, R.: Defect Diff. Forum194±199 (2001) 1199.

8 Harrison, L.G.: Trans. Faraday Soc. 57 (1961) 597.9 RuÈsing, J.; Herzig, Chr.: Intermetallics 7 (1996) 647.

10 Suzuoka, T.: J. Phys. Soc. Japan 19 (1964) 839.11 Million, P.; Ruzickova, J.; Velisek, J.; Vrestal, J.: Mater. Sci. Eng.

50 (1995) 43.12 Divinski, S.V.; Larikov, L.N.: Defect Diff. Forum 143±147

(1997) 1469.13 Benoist, P.; Martin, G.: Thin Solid Films 25 (1975) 181.14 Murch, G.E.; Rothman, S.J.: Diff. Defect Data 42 (1985) 17.15 Belova, I.V.; Murch, G.E.: Phil. Mag. 81 (2001) 2447.16 Metsch, P.; Spit, F.H.M.; Bakker, H.: phys. stat. sol. a 93 (1986)

543.17 Hart, E.W.: Acta metall. 5 (1957) 597.18 Kaur, I.; Mishin, Y.; Gust, W.: Fundamentals of Grain and Inter-

phase Boundary Diffusion, John Wiley, Chichester (1995).19 GuÈthoff, F.; Mishin, Y.; Herzig, Chr.: Z. Metallkd. 84 (1993) 584.

(Received December 22, 2001)

Correspondence address

Prof. Dr. Christian HerzigInstitut fuÈr Materialphysik, UniversitaÈt MuÈnsterWilhelm-Klemm-Str. 10, 48149 MuÈnster, GermanyTel: +49 251 833 3573Fax: +49 251 833 8346E-mail: [email protected]



264 Z. Metallkd. 93 (2002) 4

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59 fe grain boundary diffusion in nanostructured γ-fe–ni

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