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Nanocrystallization in driven amorphous materials

S. Shukla a, D.T. Wu b, H. Ramanarayan b, D. Srolovitz c,d, R.V. Ramanujan a,⇑

a School of Materials Science and Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singaporeb Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore

c Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USAd Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA

Received 15 November 2012; received in revised form 7 February 2013; accepted 8 February 2013Available online 13 March 2013

Abstract

The nanocrystallization of mechanically milled amorphous alloys was examined both experimentally and theoretically. Mechanicalmilling induces the precipitation of nanocrystals in an initially amorphous Nd–Fe–B magnetic alloy. The effects of milling speed andduration on precipitate growth/size were investigated. Milling intensity was found to significantly affect the steady-state precipitate size.Precipitate growth kinetics and steady-state precipitate size were governed by a dynamic equilibrium between defect-enhanced diffusionalprecipitate growth and impact-induced crystal attrition. A linear decrease in steady-state precipitate size with increasing milling speedwas predicted, consistent with our experimental data. Nanocrystallization in many driven amorphous alloys can be understood usingthe kinetic model developed here.� 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Metallic glasses; Mechanical milling; Precipitation kinetics; Growth kinetics; Theory and modeling (kinetics, transport, diffusion)

1. Introduction

The excellent mechanical and magnetic properties ofnanocrystalline nanocomposites and partially crystallizedamorphous structures have led to growing interest in thecontrol of crystallization of metallic glasses (MGs) [1–4].Crystallization of MGs can be induced thermally orthrough the input of some form of external energy thathelps overcome the energetic barriers associated with crys-tallization. Partial or full crystallization usually corre-sponds to a lower energy state compared with their fullyamorphous counterparts.

Annealing of amorphous systems above the crystalliza-tion temperature is the most common method to inducecrystallization. However, it is usually difficult to controlthe nucleation and growth of the crystallization processthrough this means. On the other hand, controlled nano-crystallization is feasible via other processes. These include

mechanical processing (e.g. milling [5], nanoindentation [6]and high pressure torsional deformation [2]) and electron/ion irradiation [7]. These techniques result in the formationof nanoscale crystalline grains with well-defined morpholo-gies. Hence, the thermodynamics and kinetics of nanocrystal-lization of amorphous alloys driven by non-thermal effects isof considerable scientific and technological interest.

Melt spinning, splat quenching and condensation fromthe gas phase are well-established techniques for producingamorphous alloys. Unfortunately, these methods typicallyproduce either films or ribbons and are not readily applica-ble to the production of industrially interesting bulk mate-rials and/or complex shapes. Since mechanically milledpowders can be readily consolidated, mechanical millingof melt-spun ribbons provides an interesting opportunityto induce nanocrystallization in MGs – paving the wayfor bulk processing of nanocrystalline structures of arbi-trary shape.

Unlike thermally induced nanocrystallization, the mech-anism and kinetics of deformation-induced nanocrys-tallization are not well understood [8–13]. Why is the

1359-6454/$36.00 � 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.actamat.2013.02.012

⇑ Corresponding author. Tel.: +65 6790 4342.E-mail address: [email protected] (R.V. Ramanujan).

www.elsevier.com/locate/actamat

Available online at www.sciencedirect.com

Acta Materialia 61 (2013) 3242–3248


steady-state crystal size of nanocrystals produced duringmechanical milling (MM) confined to the nanometerrange? Why is this steady-state crystal size much smallerthan that produced via thermal annealing? In this work,we examine the growth kinetics of nanocrystals viamechanical milling of melt-spun magnetic amorphousNd10Fe85B5 ribbons. We then develop a kinetic model forthe crystallization of amorphous alloys during mechanicalmilling and parameterize using experimental data toexplain the experimental observations.

This work focuses on exchange-coupled magnetic nano-composites in the Nd–Fe–B hard magnetic alloy system.While conventional permanent magnets are commonlyrare-earth-based single phase alloys (e.g. Nd2Fe14B andSmCo5) which exhibit high intrinsic coercivity, they typi-cally possess relatively low magnetization. Moreover,diminishing supplies and escalating costs of rare earth ele-ments pose new challenges for the industrial production ofsuch magnets [14,15]. The discovery of novel exchange-coupled hard–soft nanocomposites offers a promisingapproach to circumvent these limitations [16]. Rather thantrying to simultaneously enhance magnetization and coer-civity of a single phase alloy, this can be accomplished ina nanocomposite where the soft and hard magnetic proper-ties are carried by different phases. One phase is magneti-cally hard with large anisotropy constants and highcoercivity (e.g. Nd2Fe14B), while the second is magneticallysoft with large saturation magnetization (e.g. a-Fe). Previ-ous work has shown that the magnetic properties of suchnanocomposites are sensitive to the size of the magneticallysoft a-Fe crystals [16,17]; large energy products can only beobtained when the a-Fe crystals are on the nanoscale.

Controlled nanocrystallization via mechanical milling(MM) of Nd–Fe–B ribbons provides a promising methodto manufacture such high performance bulk magneticnanocomposites. Hence, we study the mechanical milling-induced precipitation of the crystalline a-Fe phase in anamorphous matrix. This study provides an understandingof the atomistic processes during mechanically inducedcrystallization of amorphous alloys and the role that theseprocesses play in determining steady-state crystal size.

2. Experimental procedure

Amorphous Nd10Fe85B5 ribbons were synthesized byinduction melting of cast buttons in a boron nitride cruci-ble in an argon atmosphere. The molten material wasejected onto a copper wheel rotating at 70 Hz with an ejec-tion pressure of 500 mbar. As-spun ribbons were crushedand milled in a Pulverisette-5 planetary ball mill in argon[18]. Crystallization kinetics were determined for millingspeeds of 150, 200, 250 and 300 rpm for several differentmilling times (intensities). The milling was stopped for5 min periods after every 10 min of milling and the millingvial was water-cooled in order to control the temperature.All pre- and post-processing of powders was performed inargon in a glove box. X-ray diffraction (XRD) data were

collected using a Bruker D8 X-ray diffractometer (Cu Karadiation) and analyzed using Rietveld refinement via theTOPAS 3.1 program; a pseudo-Voigt function was usedto model the experimental diffraction peaks. Cerium oxide(SRM 674 b, NIST) was used as an internal standard forquantitative phase analysis. TEM investigations were per-formed using a JEOL-2100F transmission electronmicroscope.

3. Results and discussion

The X-ray diffractograms of as-spun amorphous Nd10

Fe85B5 ribbons milled at 150, 200, 250 and 300 rpm for sev-eral time periods are shown in Figs. 1–4, respectively. Thefull width at half maximum (FWHM) of all the a-Fe dif-fraction peaks decreased while the intensity increased withincreasing milling time. This indicates the precipitation ofa-Fe nanocrystals within the amorphous matrix followedby an increase in crystal size. The FWHM of each of thea-Fe diffraction peaks in the samples milled at lower mill-ing speeds (Figs. 1 and 2) were much lower than in the sam-ples milled at higher speeds (Figs. 3 and 4). These resultssuggest that the milling-induced increase in crystal sizewas more pronounced for samples milled at lower millingspeeds. Consistent with the XRD data, TEM micrographsof ribbons milled for 12 h (at 300 rpm) showed that a-Fenanocrystals were dispersed in the amorphous matrix(Fig. 5).

Fig. 6 shows the a-Fe crystal size (determined by Riet-veld refinement) as a function of milling speed. The crystalsize increases with milling time and subsequently saturatesat late times. Increasing milling speed results in saturata-tion at earlier times. At 300 rpm, saturation is evidentwithin 18 h of milling, while more than 70 h are requiredat 150 rpm. Importantly, milling speed was found to signif-icantly influence crystal size; increasing milling speedresulted in smaller a-Fe nanocrystals.

Fig. 1. Rietveld refinement analyses of X-ray diffractograms of Nd10Fe85

B5 ribbons milled at 150 rpm for (a) 24, (b) 30, (c) 36, (d) 45, (e) 57, (f) 72and (g) 87 h. The symbols identify major peaks associated with crystallinea-Fe and CeO2.

S. Shukla et al. / Acta Materialia 61 (2013) 3242–3248 3243


Since amorphous alloys are not in thermodynamic equi-librium, partial crystallization of the amorphous phase canoccur if the energy input from mechanical deformation isable to overcome the activation barriers associated withnucleation and growth. Mechanical work supplied via mill-ing provides the driving force required for nucleation; sub-sequent crystal growth provides a path towardsequilibrium. We analyze the results assuming that crystalgrowth is diffusion-controlled and that atomic diffusion isenhanced by defect formation.

We recently reported that deformation induces localdensity fluctuations in the amorphous Nd–Fe–B ribbons,leading to the formation of free volume–anti-free volume(FV–AFV) pair defects during mechanical milling [8].The dilated regions have low atomic density (FV regions),while compressed regions have high atomic density (AFVregions), analogous to Frenkel (vacancy–interstitial) pairproduction in crystals under irradiation [19,20]. TheseFV and AFV regions migrate with continued milling,

Fig. 2. Rietveld refinement analyses of X-ray diffractograms of Nd10Fe85-

B5 ribbons milled at 200 rpm for (a) 12, (b) 18, (c) 24, (d) 30, (e) 36, (f) 48and (g) 63 h.


B5 ribbons milled at 250 rpm for (a) 3, (b) 6, (c) 9, (d) 12, (e) 18, (f) 24, (g)30 and (h) 36 h.


B5 ribbons milled at 300 rpm for (a) 3, (b) 6, (c) 9, (d) 12, (e) 18, (f) 24 and(g) 30 h.

Fig. 5. TEM micrograph of (a) a melt spun ribbon and (b) ribbon milledfor 180 min at 300 rpm with a-Fe nanocrystals dispersed in the amorphousmatrix.

3244 S. Shukla et al. / Acta Materialia 61 (2013) 3242–3248


facilitating the atomic redistribution required for diffusion-mediated precipitation.

While crystal growth can be understood on the basis ofFV–AFV migration, the saturation of crystal size cannotbe explained by this mechanism. Martin [21,22] suggestedthat the composition and crystal structure of the phasesformed during non-equilibrium processes (e.g. mechanicalmilling and ion irradiation) is the result of two competingprocesses: ballistic atomic jumps occurring associated withmilling/ irradiation and equilibrating atomic jumps occur-ring due to thermal diffusion. Krasnochtchekov et al.[23], Enrique et al. [24] and Bellon et al. [25] used kineticMonte Carlo simulations to show that this concept maybe extended to a variety of driven systems in order to pre-dict precipitate size, composition, stability and morphologyat various length scales.

As described above, the crystal size increases withmechanical milling time, followed by saturation (Fig. 6).This suggests that there is a competition between the twoatomistic processes. The first is FV–AFV-induced diffu-sion, which facilitates crystal growth, and the (second)competing process is impact-induced crystal attrition,which tends to lead to smaller crystal sizes.

Thus, if we consider a crystal/amorphous interface, twotypes of atomic jumps compete:

1. Atomic jumps assisted by FV–AFV migration: Fe atomsfrom the amorphous matrix jump from the amorphousmatrix into the crystal, leading to crystal growth.

2. Atomic jumps due to impacts suffered during milling: Featoms from the crystal undergo ballistic jumps from thecrystal into the adjoining amorphous matrix, resulting incrystal size reduction.

Crystal growth occurs as long as the diffusional jumpprocesses dominate the ballistic jump processes. Whenthe rate of ballistic atomic jumps becomes equal to the rateof diffusional jumps, a steady-state crystal size is achieved.The steady-state crystal size is a dynamic steady state asso-ciated with these competing atomistic effects.

Since crystal growth occurs due to FV–AFV migration,we develop an expression for the crystal growth rate interms of defect-enhanced diffusion. Also, using the analogybetween milling-induced crystal attrition and shrinking ofembryos in thermally induced crystallization, we proposean ansatz for the rate of crystal size reduction. Finally,we combine these analyses of crystal growth and crystalsize reduction to determine the rate of crystal sizeevolution.

3.1. Rate of crystal growth

We use an approach similar to that employed todescribe defect evolution in irradiated materials [26].Assuming that defect creation and annihilation occurhomogeneously, the time derivative of the net defect con-centration resulting from milling-induced deformation cf

can be written as [27,28]:

Fig. 6. Comparison of the experimental data with theoretical prediction of crystal size as a function of milling time of Nd10Fe85B5 ribbons milled at (a)150, (b) 200, (c) 250 and (d) 300 rpm.



dcf

dt¼ dc

dt� cf � c0

f

� �k2

v

h i2

k2 ð1Þ

where dc/dt is the rate of atomic defect generation per unitvolume, k2 is a rate constant, c0

f is the equilibrium defectconcentration in the ribbon before milling and kv is thestrength of available sinks. We assume bimolecular kineticsfor defect annihilation (rather than unimolecular kinetics)[29], since the dominant defects created during the mechan-ical milling of Nd10Fe85B5 ribbons are FV–AFV pairs,rather than only FV regions [8].

At steady state, the time derivative of the defect concen-tration dcf/dt = 0. Hence, setting the time derivative in Eq.(1) to zero, we obtain the steady-state atomic defect con-centration css

f .If c0

f is much smaller than cssf ,

cssf ¼

1ffiffiffiffiffik2

pk2

v

ffiffiffiffiffidcdt

rð2Þ

Atzmon et al. [30] applied defect evolution models, orig-inally developed for irradiated materials, to deformedamorphous alloys, and showed that the rate of atomicdefect generation during deformation is proportional tothe deformation rate. Assuming that the rate of deforma-tion scales with milling intensity g, we write dc/dt = kmg,where km is a constant. Substituting this expression intoEq. (2) yields

cssf ¼

1

k2v

ffiffiffiffiffiffiffiffikmgk2

rð3Þ

According to the theory of defect evolution in irradiatedalloys, the sink strength kv is related to the instantaneouscrystal size D as k2

v ¼ k3D�n, where k3 is a constant and n

is a positive number which depends on the morphologyand distribution of sinks [27,29,30]. If the amorphous/crys-tal interfaces are the active sinks and the crystallites areequiaxed, k2

v ¼ k3D�2 [28]. Inserting this expression intoEq. (3) yields

cssf ¼

D2

k3


rð4Þ

The defect enhanced diffusion coefficient DDED can bewritten in terms of the steady-state atomic defect concen-tration css

f as [28]:

DDED ¼ D0cssf ¼ D0

D2

k3


rð5Þ

where D0 is a constant. The rate of crystal growth can nowbe expressed as [28,29]:

dDdt¼ k0DDED

Dð6Þ

where k0 is proportionality constant. Combining Eqs. (5)and (6), we can now write

dDdt¼

k0D0D2ffiffiffiffiffiffikmgk2

qDk3

¼ k0D0

Dk3


r

or

dDdt¼ kgD ð7Þ

where kg is a crystal growth parameter:

kg ¼ k0

D0

k3


rð8Þ

3.2. Rate of crystal attrition

We make an analogy between the shrinkage of embryosduring crystallization and impact-induced crystal attritionduring milling to develop an expression for the rate of crys-tal size reduction. We assume that impact forces duringmilling can force Fe atoms to jump across the crystal–amorphous matrix interface, resulting in a reduction incrystal size. Classical nucleation theory [31] assumes thatthe rate of shrinkage of a cluster is proportional to its inter-facial area pD2, where D is the cluster size. Analogously,the attrition rate of a crystal during milling-induced crys-tallization should also be proportional to the area of thecrystal–amorphous interface, pD2. Moreover, as theshrinkage rate of embryos in classical nucleation theorydepends on thermal activation, we assume that the attritionrate of the crystals depends on the intensity of the mechan-ical milling g. This assumption is consistent with Martin[22], who suggested that the ballistic jump frequencyshould depend on milling intensity in the same way thatthe diffusional jump frequency depends on temperature.Thus, we make the following ansatz for milling-inducedcrystal attrition:

dDdt¼ �kngD2 ð9Þ

where kn is constant with respect to g and t. We define acrystal attrition parameter ka as

ka ¼ kng ð10ÞThus, the crystal attrition rate can be expressed as

dDdt¼ �kaD2 ð11Þ

3.3. Rate of crystal size evolution

Combining the expressions for the rates of crystalgrowth (Eq. (7)) and attrition (Eq. (11)), the overall crystalsize evolution is

dDdt¼ kgD� kaD2 ð12Þ

In Eq. (12) we have related the rate of crystal growth,crystal attrition and the steady-state crystal size to the mill-ing intensity g (recall that ka is proportional to g and kg tog1/2). The first and second terms in Eq. (12) represent crys-tal growth and attrition, respectively. Eq. (12) can besolved for the time-dependent crystal size D as:



DðtÞ ¼ ðkg=kaÞ1þ ðkg=kaÞ

D1� 1

n oexpð�kgtÞ

ð13Þ

We compare the experimental measurements of crystalsize vs. time at various milling speeds to our model. Thepredicted crystal sizes (Eq. (13)) are shown together withthe experimental data in Fig. 6. In making this comparison,we fitted the unknown constants, D1, ka and kg to theexperimental data to obtain the best fit. The values of ka

and kg, determined in this manner, are shown in Table 1.The theoretical results and the experimental data are invery good agreement (see Fig. 6). Since the crystal size evo-lution rate is zero at steady-state, we determine the steady-state crystal size Dss by setting Eq. (12) to zero:

Dss ¼kg

kað14Þ

The same result can be found from Eq. (13) by going tothe infinite-time limit.

While Fig. 6 shows a good match between experimentalcrystal size evolution data and our model, it does notexplain why the steady-state crystal size changes with mill-ing speed in the way that it does. To understand this, weneed to determine how the crystal growth parameter kg

and the crystal attrition parameter ka vary with millingspeed (Eq. (14)). Milling intensity g is defined as the rateof momentum transfer per unit mass of powder particlesduring mechanical milling and can be expressed as[18,32,33]:

g ¼ MbV vfMp

ð15Þ

where Vv is the linear velocity of the vial, Mb is the mass ofthe milling balls, Mp is the mass of the powders and f is theimpact frequency. The impact frequency and linear velocityof the vial can be written in terms of the rotational speedsof the frame xp and vial xv [18,21] as f = k(xp� xv) andVv = kb(xp� xv)Rb, respectively. k and kb are constantsand Rb is the radius of the frame. Substituting these expres-sions in Eq. (15) yields an expression for the milling intensity:

g ¼ MbkbRbkðxp � xvÞ2

Mpð16Þ

In the Pulverisette-5 ball mill, xp and xv are related byxv = �1.25xp [18] and the ratio of the mass of the ballsto the mass of the powder is Mb/Mp = 20. Hence, for ourexperimental system

g ¼ 101:25kkbRbx2p ð17Þ

Substituting this expression for the milling intensity ginto Eqs. (8) and (10), we find:

kg ¼ k0

D0

k3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi101:25kkbRbkm

k2

sxp ð18Þ

and

ka ¼ ½101:25knkkbRb�x2p ð19Þ

Eqs. (18) and (19) suggest that the rate of increase of thecrystal attrition parameter dka/dxp is greater than dkg/dxp

for all milling speeds xp. To compare this prediction withthe experimental data, we used the values of kg and ka

determined from the fitting to the experimental data (seeFig. 6 and Table 1) and plotted these parameters withrespect to milling speed xp (Fig. 7). We observed that theexperimentally observed dependence of kg and ka inFig. 7 is consistent with the theoretical model: kg / xp

(Eq. (18)) and ka / x2p (Eq. (19)).

Since Dss ¼ kg

ka(Eq. (14)) and kg and ka are proportional

to xp and x2p, respectively; Dss should be proportional to

1/xp. We plotted Dss vs. 1/xp to compare our experimentaldata with our prediction (Fig. 8). The experimental datacould be fit to a straight line with a positive slope, consis-tent with the model. The data point for the lowest millingspeed, i.e. 150 rpm, deviates slightly from the linear fit.

At low milling speeds powder particles tend to stick tothe ball surfaces and the milling container. When materialgets stuck to the milling media surfaces, ball-to-powderratio (BPR) increases artificially (the stuck powder parti-cles do not participate in the milling). This artificialincrease in the ball-to-powder ratio and the fact that muchless powder is available for milling, lead to an increasedimpact frequency and hence milling intensity. Thus, whencold welding is significant, powder particles experience ahigher milling intensity than the milling intensity they expe-rience when cold welding is minimal and hence the correla-tion between Dss vs. 1/xp deviates slightly from a straightline when mechanical milling is performed at low millingspeeds, e.g. 150 rpm.

4. Conclusions

Amorphous Nd10Fe85B5 ribbons were synthesized bymelt spinning and these ribbons were subsequently sub-jected to mechanical milling. This mechanical work leadsto deformation-induced nucleation and growth of a-Fenanocrystals in the amorphous matrix. The crystal growthkinetics and the evolution of grain size were investigated byRietveld refinement XRD analysis. A new kinetic modelwas proposed to explain the experimental findings. Themain conclusions of this study are as follows:

� Milling speed significantly affects steady-state crystalsize. Increasing milling speed resulted in decreasedsteady-state crystal a-Fe nanocrystals size.

Table 1Variation of ka and kg obtained by fitting the theoretical expressions to theexperimental data of crystal size as a function of milling time forNd10Fe85B5 ribbons (see Fig. 6).

150 rpm 200 rpm 250 rpm 300 rpm

ka (min�1) 0.012 0.022 0.034 0.049kg (nm3 min�1) 0.142 0.231 0.309 0.387



� The kinetic model predicted steady-state crystal size rep-resents a balance between two competing atomistic pro-cesses: defect-enhanced diffusional atomic jumps(leading to crystal growth) and impact-induced ballisticatomic jumps (leading to crystal attrition).� According to the model, the rates of crystal attrition and

crystal growth were a strong function of milling speed.With increasing milling speed, the increase in the rateof crystal attrition was predicted to be quadratic andgreater than the linear increase in the rate of crystalgrowth. The model also predicted a linear decrease insteady-state crystal size with respect to increasing mill-ing speed. These predictions are consistent with theexperimental results.� The kinetic model developed in this work is applicable

to crystallization in a variety of systems which are drivenfar from equilibrium by external processes such asmechanical milling and irradiation.

References

[1] Valiev RZ. J Mater Sci 2007;42:1483.

[2] Li W, Li L, Nan Y, Xu Z, Zhang X, Popov AG, et al. J Appl Phys2008;104:023912.

[3] McHenry ME, Johnson F, Okumura H, Ohkubo T, Ramanan VRV,Laughlin DE. Scripta Mater 2003;48:881.

[4] Lee SW, Huh MY, Fleury E, Lee JC. Acta Mater 2006;54:349.[5] Choi PP, Povstugar IV, Kwon YS, Yelsukov EP, Kim JS. Mater Sci

Eng A 2007;448–451:1079.[6] Kim JJ, Choi Y, Suresh S, Argon AS. Science 2002;295:654.[7] Nino A, Nagase T, Umakoshi Y. Mater Sci Eng A 2007;448–

451:1115.[8] Shukla S, Banas A, Ramanujan R. Phys Status Solidi RRL 2011;5–

6:169.[9] Kohda M, Haruyama O, Ohkubo T, Egami T. Phys Rev B: Condens

Matter Mater Phys 2010;81:092203.[10] Lewandowski JJ, Greer AL. Nat Mater 2006;5:15.[11] Jiang WH, Atzmon M. Scripta Mater 2006;54:333.[12] Chen H, He Y, Shiflet GJ, Poon SJ. Nature 1994;367:541.[13] Eckert J. Nanostruct Mater 1995;6:413.[14] Long K, Gosen B, Foley N, Cordier D. In: Sinding-Larsen R,

Wellmer FW, editors. Non-renewable resource issues. Heidel-berg: Springer; 2012.

[15] Du X, Graedel TE. Environ Sci Technol 2011;45:4096.[16] Betancourt I, Davies HA. Mater Sci Technol 2010;26:5.[17] Kneller EF, Hawig R. IEEE Trans Magn 1991;27:3588.[18] Suryanarayana C. Prog Mater Sci 2001;46:1.[19] Srolovitz D, Vitek V, Egami T. Acta Metall 1983;31:335.[20] Egami T, Srolovitz D. J Phys F 1982;12:2141.[21] Martin G, Gaffet E. J Phys Colloques 1990;51:4.[22] Martin G. Curr Opin Solid State Mater Sci 1998;3:552.[23] Krasnochtchekov P, Averback RS, Bellon P. JOM 2007;59:46.[24] Enrique RA, Bellon P. Phys Rev Lett 2000;84:85.[25] Bellon P, Martin G. In: Pfeiler W, editor. Alloy physics: a

comprehensive reference. Weinheim: Wiley-VCH; 2007.[26] Park B, Spaepen F, Poate JM, Jacobson DC, Priolo F. J Appl Phys

1990;68:4556.[27] Jiang WH, Atzmon M. J Mater Res 2003;18:755.[28] Biloni H, Boettinger WJ. In: Cahn RW, Haasen P, editors. Physical

metallurgy. Amsterdam: Elsevier Science; 1996.[29] Woo CH. In: Yip S, editor. Handbook of materials modelling. Ber-

lin: Springer; 2005.[30] Jiang WH, Pinkerton FE, Atzmon M. J Appl Phys 2003;93:9287.[31] Brent Fultz JH. In: Pfeiler W, editor. Alloy physics: a comprehensive

reference. Weinheim: Wiley-VCH; 2007.[32] Tian HH, Atzmon M. Acta Mater 1999;47:1255.[33] Kwon YS, Kim JS, Kim JC, Kwon YJ, Povstugar I, Yelsukov E,

et al. J Nanosci Nanotechnol 2010;10:336.

Fig. 7. Comparison of experimental values with theoretical prediction of ka (crystal attrition parameter) and kg (crystal growth parameter) of Nd10Fe85B5

ribbons. Samples were milled at 150, 200, 250 and 300 rpm until the steady-state was achieved.

Fig. 8. Comparison of the experimental values with theoretical predictionof variation in steady-state crystal size with inverse milling speed; red lineis a linear fit to experimental data. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of thisarticle.)


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