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Scripta Materialia 60 (2009) 945–948

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On the stagnation of grain growth in nanocrystalline materials

Junjie Li, Jincheng Wang* and Gencang Yang

State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China

Received 11 December 2008; revised 26 January 2009; accepted 6 February 2009Available online 13 February 2009

The stagnation of grain growth with solute segregation in nanocrystalline materials is often analyzed by using two kinetic models[J.E. Burke, Trans. Metall. Soc. AIME 180 (1949) 73; A. Michels, C.E. Krill, H. Ehrhardt, R. Birringer, D.T. Wu, Acta Mater. 47(1999) 2143], in which it is supposed that solute drag stops grain growth. However, we show that the drag force in these kineticmodels is not equivalent to the solute drag force, but to the reduced driving force due to the decrease in grain boundary energy.� 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain growth; Segregation; Grain boundary energy; Nanocrystalline materials

Nanocrystalline materials have generated greatinterest due to their unusual mechanical, electrical, mag-netic and corrosion properties which are ascribed to thefine structural length scale. Due to the importance of thefine grain size, the thermal stability of nanocrystallinematerials with respect to grain growth is of great con-cern. Inhibiting grain coarsening by grain boundary(GB) segregation is a promising approach, and has beenwidely demonstrated in various nanocrystalline materi-als [1–10]. The mechanism of the improved stabilitydue to solute or impurity segregation, however, is stillin doubt. Two kinds of models have been proposed: akinetic one in which solute drag stops grain growth,and a thermodynamic one in which the driving force,i.e. GB energy, is suppressed. Both of the models havebeen successfully employed to account for some experi-mental results. It is commonly recognized that these twokinds of models are different from each other, and as faras we know no relation between them has been estab-lished. However, in this paper we show that the kineticmodels proposed by Burke [11] and later by Grey andHiggins [12], and further developed by Michels et al.[13], can be related to the thermodynamic change, i.e.the suppression of the GB energy due to solute segrega-tion. It is found that the drag force in these kinetic mod-els can be equated with the reduced driving force due tothe decrease in GB energy.

It is well known that the presence of solute or impu-rity segregation in GB regions may give rise to a signif-icant drag force on GB migration. Lucke and Detert [14]

1359-6462/$ - see front matter � 2009 Acta Materialia Inc. Published by Eldoi:10.1016/j.scriptamat.2009.02.015

* Corresponding author. Tel.: +86 29 88460650; fax: +86 2988491484; e-mail: jchwang@nwpu.edu.cn

proposed the first quantitative model describing thisphenomenon. Cahn [15] and Lucke and Stuwe [16] ex-tended this model by considering that the interaction en-ergy between the solute atoms and the GB varies withthe distance of an atom from the middle of the GB.By correlating the solute concentration to the interac-tion energy, the steady-state composition profile acrossthe GB migrating with a constant speed was determinedto calculate the solute drag force. Based on the evalua-tion of the free energy dissipation, Hillert and Sundman[17] presented a more general formulation which allowsfor the treatment of phase interfaces as well as GBs.They demonstrated that their treatment is equivalentto Cahn’s treatment for GBs in dilute solutions. To re-late the GB velocity vGB to the driving force F, the equa-tion suggested by Cahn can be written as:

F ¼ vGB

MGB

þ eX gvGB

1þ x2v2GB

; ð1Þ

where MGB is the mobility of the GB in the pure metal,i.e. the so-called intrinsic mobility, Xg is the atom frac-tion of solute in the bulk, and e and x are two param-eters of the model that depend both on the interactionenergy between the solute atom and the GB and on thediffusivity of the solute in the vicinity of the GB. InCahn’s model, it can be found that at the low velocityextreme the apparent mobility of GB motion with sol-ute drag is lower than that in the pure metal (MGB)and the GB velocity is decreased. However, in this sit-uation the linear relation between the driving force andthe GB velocity still holds, which means a parabolicgrain-growth law should also be valid. The solute dragmodel of Cahn does not provide an explanation for a

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946 J. Li et al. / Scripta Materialia 60 (2009) 945–948

steady-state grain size. As far as we know, the experi-mental observed stagnation of grain growth in noncrys-talline materials has never been directly analyzed byusing Cahn’s model.

The most widely used kinetic model for the descrip-tion of inhibited grain growth is the one proposed byBurke [11]. In his model the kinetic equation for graingrowth under ideal conditions is modified by incorporat-ing a drag force influence, yielding:

dDdt¼ A

1

D� 1

Dmax

� �¼ aMr0

1

D� 1

Dmax

� �; ð2Þ

where D is the mean grain diameter after an annealingtime t, A is a temperature-dependent rate constantwhich is proportional to the product of GB mobilityM and GB energy r0, a is a constant of proportionalityand Dmax is the maximum grain size resulting from thedrag force. It should be noted that the drag force term,f = A/Dmax, in Eq. (2) is independent of grain size andGB velocity. Burke’s model was originally proposed toaccount for the GB pining by second-phase particles(Zener drag) or the sample surface (thickness effect).Grey and Higgins [12] first applied this model to thecase of GB pinning by the segregation of solute orimpurity (solute drag). Since then, this model has beenwidely used to analyze the stagnation of grain growth(e.g. [2–4]). Michels et al. [13] noticed that during graingrowth in nanocrystalline materials the solute concen-tration at the GB increases linearly with the averagegrain radius, resulting in an increased solute drag forceaccording to the solute drag theory of Cahn. There-fore, they modified Burke’s model to the followingform:

dDdt¼ A

1

D� D

D2max

� �¼ aMr0

1

D� D

D2max

� �; ð3Þ

where the term AD=D2max represents the grain-size-depen-

dent drag force owing to solute segregation in the GB.This modified model also has been extensively employedto analyze grain-growth kinetics (see Refs. [5–7]).

However, the drag force terms in the above two mod-els are velocity independent, and the physical meaningsof these velocity-independent components are still un-clear. Grey and Higgins [12] assumed that a kind of sol-ute cluster may account for the velocity-independentdrag force. But this kind of solute cluster has never beenobserved in practical experiments. According to the sol-ute drag theory of Cahn, in an equilibrium state wherethe GB is not moving, the GB segregation gives rise tothe absence of net force at the interface. When a GB ismoving under a driving force, the concentration profilefalls behind the GB position, and the resultant brokensymmetry between the concentration profile and theGB position brings a net attractive force, i.e. solute dragforce, between them. The solute drag force is propor-tional not only to the excess of solute atoms at theGB, but also to the GB velocity. So there is no solutedrag force when the GB velocity is zero, which is incon-sistent with the behavior of the velocity-independentdrag force. Therefore, the drag force considered in themodel of Burke and the model of Michels et al. cannotbe interpreted as the solute drag force as proposed by

Cahn. The physical meaning of the velocity-independentterms, A/Dmax and AD=D2

max; is still unclear.Moreover, in Cahn’s model, due to the influence of

the velocity dependence of the solute drag force, fromEq. (1) it can be obtained that the apparent GB mobilityin the low velocity limit, Mapp, can be related with theintrinsic GB mobility MGB and the solute-drag mobilityMdrag as:

1

Mapp

¼ 1

MGB

þ 1

Mdrag

: ð4Þ

So in the theory of Cahn it can be claimed that theactual mobility of GB motion with solute segregationdecreases due to the solute drag effect. But in the modelof Burke or the model of Michels et al. the drag force isvelocity independent. So from Eqs. (2) and (3) it can beseen that the GB mobility, which is proportional to thecoefficient A, is not changed owing to the drag effect.The decrease in GB velocity in these two models iscaused by the decrease in the driving force, but not bythe decrease in the GB mobility. Rabkin [18] has shownthat if the drag force in the model of Michels et al. ismodified to correlate with the GB velocity as the solutedrag force in Cahn’s model, their model will show nolimiting grain size, which means the stagnation of graingrowth cannot be reproduced by this model. Therefore,it is not reasonable to take the velocity-independentdrag force in the model of Burke and the model of Mich-els et al. as the solute drag force.

What are the really physical meaning of the dragforce terms in the model of Burke and the model ofMichels et al.? We may get some inspiration from thethermodynamic strategy for improving the thermal sta-bility of nanocrystalline materials. The idea of thermo-dynamically stabilizing nanocrystalline materials bysuppressing the GB energy was first introduced byWeissmuller [19] who has given a theoretical descriptionand shown that a minimum could indeed exist on thefree energy curve in the nanocrystalline region. Addi-tional analytical models have been further developedby Kirchheim et al. [20,21], Krill et al. [8,22] and Bekeet al. [23,24]. According to these theories a metastableequilibrium with a certain grain size can be attainedwhen the GB energy r reduces to zero. Some reasonableagreements between the predictions of thermodynamicanalysis and the experimental observations have beenshown in Refs. [20–21,25–26].

According to the thermodynamic treatments ofWeissmuller [19] and Kirchheim [20], the reduced GBenergy due to solute segregation can be written as:

r ¼ r0 � CðDH seg þ RT ln X gÞ; ð5Þwhere r0 is the grain boundary energy for the pure sol-vent, Xg is the bulk concentration, DHseg is the enthalpyof segregation and C is the solute excess at the GB. Thethermodynamic treatments only concentrate on the finalstate of the system and it is commonly assumed that theGB is finally saturated with solute atoms, i.e. C = C0.After making this assumption and combining with thesolute conversion law in a closed system

X total ¼3CV M

Dþ X g ð6Þ

J. Li et al. / Scripta Materialia 60 (2009) 945–948 947

Kirchheim [20] has obtained that the final grain sizeDmax at the metastable equilibrium state could be givenimplicitly as:

r0 ¼ C0 RT ln X total �3C0V M

Dmax

� �� H seg

� �; ð7Þ

where VM is the molar volume of the alloy.In all above thermodynamic treatments the kinetic

process is omitted. If we try to incorporate the GB en-ergy decrease into the ideal grain-growth law, variationof C with the grain diameter should be considered in-stead of assuming a saturation state. After insertingEq. (5), which describes the variation of GB energy withthe solute segregation, into the ideal grain-growth law,we obtained that:

dDdt¼ aM r0 � C DH seg þ RT ln X g

� �� 1

D: ð8Þ

It can be seen that the modified ideal grain-growthmodel is related to the variation in the solute excess atthe GB. With increasing average grain size, fewer GB sitesare left available for the segregating atoms, which shouldthen be thrown into the bulk phase. But those dissolvedsolute atoms may be captured again by moving GBs be-fore GBs are saturated with solute atoms. The occurrenceof solute entrapment by moving GBs results in an increasein the solute excess at GBs. The changing manner of thesolute excess with grain size will influence the kinetics ofgrain growth. In the following, two ways in which the sol-ute excess at the GB changes are assumed.

Firstly, we assume the solute excess at the GB C in-creases linearly with the average grain size D. In fact,this linear relation has been observed in nanocrystallinePd81Zr19 [8]. According to Eq. (6), we can find:

C ¼ ðX total � X gÞ3V m

D: ð9Þ

So if C changes linearly with D, the bulk concentrationXg should hold a constant value. In fact, during isother-mal grain growth of nanocrystalline Fe75Si25 alloy, it hasbeen observed that the lattice parameter changes rapidlyat the beginning and then maintains an almost constantvalue at the time of highest grain growth [6]. Therefore,the assumed linear relation between C and D may bereasonable for practical process. Inserting Eq. (9) intoEq. (5), we obtain the relation between the GB energyand the average grain size as:

r ¼ r0 � DðX total � X gÞ

3V mðDH seg þ RT ln X gÞ: ð10Þ

Then the final grain size Dmax at the metastable equilib-rium state (r = 0) can be obtained as:

Dmax ¼3V mr0

ðX total � X gÞðDH seg þ RT ln X gÞ: ð11Þ

Inserting Eqs. (9) and (11) into Eq. (8) gives:

dDdt¼ aM r0�D

ðX total�X gÞ3V m

ðDH segþRT lnX gÞ� �

1

D

¼ aM r0�Dr0

Dmax

� �1

D¼ aMr0

1

D� 1

Dmax

� �; ð12Þ

which is just the same form as Eq. (2) proposed byBurke [11].

Secondly, we assume the solute excess at the GB isproportional to D2, i.e. C = bD2, where b is a constant.Then, Eq. (5) can be rewritten as:

r ¼ r0 � bD2 DH seg þ RT ln X g

� �: ð13Þ

In this situation the final grain size Dmax at the metasta-ble equilibrium state (r = 0) can be obtained as:

D2max ¼

r0

b H seg þ RT ln X g

� � : ð14Þ

Inserting Eq. (14) and the relationship C = bD2 into Eq.(8) gives:

dDdt¼ aM r0 � bD2 DH seg þ RT ln X g

� �� 1

D

¼ aM r0 � D2 r0

D2max

� �1

D¼ aMr0

1

D� D

D2max

� �; ð15Þ

which is just the same form as Eq. (3) proposed byMichels et al. [13].

It can now be seen that the velocity-independent dragforce terms in the model of Burke and the model ofMichels et al. are equivalent to the reduced driving forcedue to the decrease in GB energy. We are not surewhether the physical origin of the velocity-independentdrag force is really due to the decrease in GB energy;however, this is at least a possible interpretation. Itshould be noted that although the model of Michelset al. was proposed in order to improve the model ofBurke, the model of Burke is more reasonable than thatimproved by Michels et al. from the viewpoint of GB en-ergy decreasing due to solute segregation, because theassumption that the solute excess at the GB increaseslinearly with the average grain size has been confirmedby experimental observations. However, just as demon-strated by Michels et al. [13], a simple rescaling of themobility parameter is sufficient to map the solution forD(t) of their model onto Burke’s solution, and the acti-vation energies Ea for GB motion calculated from thetemperature dependence of the mobility parameter areidentical for both models. Hence, for the analyses ofgrain size evolution with the annealing time, both mod-els are suitable.

According to the way in which a metastable equilib-rium state (r = 0) is attained, we can divide the graingrowth into three cases: (1) grain growth stops beforethe GBs are saturated with solute atoms, i.e. C < C0;(2) grain growth stops just as the GBs are saturated withsolute atoms; (3) grain growth is continuous after theGBs are saturated with solute atoms and stops after acertain time. For the first case, the maximum grain sizeDmax can be obtained by Eq. (11), and the grain-growthprocess can be described by Eq. (12), which is equivalentto Burke’s model. For the third case, the presuppositionthat the solute excess at the GB increases continuouslywith the average grain size is not satisfied, so Eq. (11)cannot be used to obtain the maximum grain size andEq. (12) is also not suitable to describe the grain-growthprocess, while Eq. (7) proposed by Kirchheim [20]should be used to calculate Dmax. For the second case,the following relationship is satisfied:

948 J. Li et al. / Scripta Materialia 60 (2009) 945–948

C0 ¼X total � X g

3V mDmax: ð16Þ

Inserting Eqs. (16) into (11) will give an implicit equa-tion for Dmax, which is just the same to Eq. (7) proposedby Kirchheim [20].

Kirchheim [20] has obtained that the temperaturedependence of Dmax can be derived by differentiatingEq. (7) with respect to T as:

dð1=DmaxÞd ln T

¼ X g ln X g

3C0V m: ð17Þ

Just as shown by Kirchheim et al. [20–21,25], althoughthe right-hand side of Eq. (17) depends on temperaturetoo, this equation turns out to be useful for a compari-son with experimental results if variation of Xg withtemperature is negligible. Whereas, if the grain growthstops before the GBs are saturated with solute atoms,according to Eq. (11) the temperature dependence ofDmax can be derived as:

dð1=DmaxÞdT

¼ ðX total � X gÞR ln X g

3V mr0

: ð18Þ

Hence, it should be possible to estimate whether or not theGBs are saturated with solute atoms by detecting the tem-perature dependence of Dmax. However, our calculationsshow that for a certain range of temperature the differencebetween Eq. (17) and (18) is insignificant. The experimen-tal results for nanocrystalline RuAl [5], which have beenanalyzed by Kirchheim [20], are redrawn both in a D�1

maxvs. T plot and a D�1

max vs. lnT plot. We find that a linearfit is reasonable for both plots. The standard deviationsof the linear fit and the correlation coefficients of the datafor the two cases are comparable. Therefore, due to thelimited experimental data it is difficult to estimate whetheror not the GBs are saturated with solute atoms by detect-ing the temperature dependence of Dmax. In order toascertain the actual variation process of solute segrega-tion at the GB, more specific detection of the variationof solute concentration in the bulk phase and the GBshould be conducted.

Our analysis has shown that the velocity-independentdrag forces considered by Burke and Michels et al. areequivalent to the reduced driving forces due to the de-crease in GB energy. In fact, not only the GB energybut also the mobility of the GB are altered by the pres-ence of solute segregation. According to the solute dragtheory of Cahn, the apparent GB mobility Mapp writtenas M in the model of Burke (Eq. (2)) and the model ofMichels et al. (Eq. (3)), can be written as Mapp = 1/(1/MGB + 1/Mdrag). Due to the poor knowledge of theintrinsic GB mobility MGB and the solute-drag mobilityMdrag, it is only possible to take Mapp as a fitting param-eter. Because the apparent mobility is a mixture of theintrinsic mobility and the solute-drag mobility, theapparent activation energy obtained by fitting the exper-imental data may not be a constant at a certain temper-ature range, which has been observed in experimentswith nanocrystalline RuAl alloys [5]. Moreover, a recentstudy by Ames et al. [27] has shown that due to the effectof the microstrain the intrinsic mobility is also depen-

dent on the grain size, which results in the functionalform of the grain-growth law revealing a positive curva-ture (concave upward). However, the concave-upwardcurvature has never been observed in systems exhibitingsolute or impurity segregation, for the effects of GB seg-regation on the GB energy and the mobility will makethe functional form of the grain-growth law producenegative curvature. It can therefore be concluded thatthe effect of the GB segregation on the grain-growthlaw is more prominent than the effect of microstrain.

In summary, the physical meaning of the velocity-inde-pendent drag force term in the model of Burke [11] and themodel of Michels et al. [13] is discussed. We find that whenthe ideal grain-growth law is modified by incorporatingthe GB energy decrease due to solute segregation, a ki-netic grain-growth model which is equivalent to the modelof Burke or the model of Michels et al. can be obtained.This equivalence may indicate that the physical meaningof the velocity-independent drag force is a reduced drivingforce due to the decrease in GB energy.

This work has been supported by the NationalNatural Science Foundation of China (Grant No.50401013) and the Doctorate Foundation of Northwest-ern Polytechnical University.

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