disintegration of nanocrystalline metals via vacancy diffusion along grain boundaries

Computational Materials Science 76 (2013) 37–42

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A mesoscopic model of dissolution/disintegration of nanocrystalline metals via vacancy diffusion along grain boundaries L. Klinger, I. Gotman, E. Rabkin ⇑ Department of Materials Science and Engineering, Technion – Israel Institute of Technology, 32000 Haifa, Israel

a r t i c l e

i n f o

Article history: Received 26 July 2012 Received in revised form 24 October 2012 Accepted 28 October 2012 Available online 24 November 2012 Keywords: Corrosion Oxidation Grain boundary diffusion Surface diffusion Nanocrystalline metals

a b s t r a c t We propose a model describing the growth of a two-dimensional subsurface pore induced by vacancy supersaturation on the surface of a polycrystal. We demonstrate that the pore develops narrow slits propagating along the grain boundaries which may lead to disintegration of nanocrystalline metal by grain detachment. Analytic models describing initial and final (on the verge of grain detachment) stages of pore growth are proposed. The models’ predictions are in good agreement with the numerical simulation results. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction A number of processes at the surfaces and interfaces in metals and alloys can cause a local supersaturation of vacancies in the near-surface/interface layer. The examples include surface [1,2] and interfacial [3] oxidation, anodic dissolution [4,5], and Kirkendall effect during the initial stages of interdiffusion between two dissimilar materials [6]. These excess vacancies can diffuse along short circuit diffusion paths (grain boundaries (GBs) and their triple junctions, dislocation cores, etc.) and cause cavitation at some distance from the surface/interface where they were formed. In our previous work we formulated a model describing the growth kinetics and morphology of interfacial pores formed as a result of precipitation of excess vacancies [7]. We demonstrated that the pore growth can proceed even at room temperature, and can cause disintegration by grain detachment in nanocrystalline Cu [7]. Our model is based on an assumption that GBs in the near-surface layer of a material are not efficient as vacancy sinks and simply conduct the excess vacancies toward pore nucleation sites at the GB triple junctions. The decrease of GB propensity for vacancy absorption can be caused either by impurities blocking dislocation climb [8], or by normal stresses developing at the GBs [9,10]. The problem of polycrystal disintegration by grain detachment, even at a slow rate, can pose a serious challenge in the case of load-bearing nanocrystalline metal implants in human body. The detached nano-grains – nanoparticles – will be first picked up by ⇑ Corresponding author. Fax: +972 4 8295677. E-mail address: [email protected] (E. Rabkin). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.10.032

and interact with the phagocytic cells of the immune system (e.g. macrophages), which may increase the host’s susceptibility to infections and cancer and/or lead to toxicity in the host [11–13]. Metallic nanoparticles can undergo rapid corrosion in the body fluids thus increasing the level of potentially allergenic and toxic ions. Transition metals are particularly effective as catalysts of oxidative stress in cells, tissues, and biofluids [14]. The nanoparticles can be further disseminated to the liver, spleen, or lymph nodes potentially inducing granulomas or granulomatoid lesions, fibrosis and lymph node necrosis [15]. The aim of this work was to further develop our earlier proposed model [7] and to identify the factors controlling pore morphology and growth rate, and the rate of grain detachment caused thereby. We also developed a simple analytic approach allowing one to estimate the rate of grain detachment during dissolution. This approach can be useful in predicting the stability of implant materials against catastrophic disintegration. 2. The model For the sake of completeness we will provide here the details of the model presented in Ref. [7]. We consider a two-dimensional polycrystal (see Fig. 1), and a triple junction of three GBs, OA1, OA2, and OA3, as a natural location for the heterogeneous nucleation of a pore. This pore grows due to the GB vacancy flux, jGB, from the external surface to the GB-pore triple junctions (points B1, B2). In the steady-state approximation this flux can be written as

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L. Klinger et al. / Computational Materials Science 76 (2013) 37–42

Fig. 1. A two-dimensional subsurface pore growing due to the GB diffusion flux of vacancies, jGB, from the surface where they are produced (by oxidation or anodic dissolution).

jGB ¼

mGB DGB lA  lB kT

lAB

ð1Þ

;

where lA is the chemical potential of vacancies in the surface layer (this is a model parameter), lB is the chemical potential of vacancies at the GB-pore triple junction (which depends on the pore curvature), mGB is the number of mobile vacancies per unit area of the GB, DGB is GB diffusion coefficient of vacancies, lAB is the distance between the leading edge of the pore and free surface, and kT has its usual thermodynamic meaning. Eq. (1) demonstrates that pores located closer to the surface will grow faster. In the framework of our model of a triple junction pore this means that the rate of pore growth increases with decreasing grain size. Evolution of the pore shape is governed by atom diffusion along the internal surfaces of the pore (B1NB2 and B1MB2)

Vn ¼ X

@js ms Ds X @ 2 ls ; ¼ @s kT @s2

ls ¼ Xcs js :

ð2Þ

In this equation Vn is normal (from vacuum to grain) velocity of the pore internal surface, js is the diffusion flux of atoms along the surface (M ? B2 ? N), s is the coordinate (arc length) along the pore surface, ms is the number of mobile atoms per unit area of the pore surface, Ds is the surface self-diffusion coefficient, X is the atomic volume, ls is the chemical potential of the atoms on the internal surface of the pore, cs is the surface energy and js is the curvature of the pore surface:

js ¼

@h ; @s

ð3Þ

Here h is the slope of the pore surface with respect to the positive direction of x-axis. The following boundary conditions complete the diffusion problem

js jM ¼ js jN ¼ 0;

js jB2   js jB2 þ ¼ jGB ;



ls B2 ¼ lB ;

ð4aÞ ð4bÞ

sin hjN ¼ cGB =2cs ;

ð4cÞ

cosðhGB  hB2  Þ  cosðhGB  hB2 þ Þ ¼ cGB =cs ;

ð4dÞ

hjM ¼ 0;

where hGB is the GB slope with respect to the positive direction of x-axis, and cGB is the GB energy (assumed to be equal for all three GBs forming the junction). The first condition in Eq. (4a) implies that we neglect self-diffusion along the GB OA3 into the pore caused by the local curvature-related supersaturation of vacancies (Gibbs– Thomson effect). It is this flux that controls the slow process of

shrinkage of GB pores [5]. This assumption is justified when the supersaturation of vacancies on the surface (A1A2) is much higher than the curvature-related supersaturation, and, hence, jGB >> js jN . Eq. (4d) and the second condition in Eq. (4c) represent conditions of mechanical equilibrium at the triple junctions of isotropic surfaces and GBs (i.e. @ cs =@h ¼ @ cGB =@h ¼ 0). In what follows we will consider an equilibrium symmetric triple junction in which the angles between all three GBs are 2p/3. In our previous work we demonstrated that this geometry is more stable against the detachment of the middle grain A1OA2 than alternative morphologies in which some GB migration in the near-surface region has occurred [7]. The set of Eqs. (1)–(4) was solved employing the numerical scheme proposed by Dornel et al. [16]. The following dimensionless coordinates (X, Y), arc length (S), treatment time (s), interface energy (G), chemical potential (M), flux (J), and diffusivity (d) were introduced

s ¼ t=t0 ; G ¼ c=c0 ; M ¼ l=l0 ; J ¼ j=j0 ; d ¼ Dm=D0 m0 ;

X; Y; S ¼ x; y; s=l0 ;

ð5aÞ

where l0, D0m0 and c0 are arbitrarily chosen units of length, diffusivity, and interface energy, respectively, and

l0 ¼

Xc0 l0

; t0 ¼

kT

X2 m0 D0

4

l 0 ; j0 ¼

Xm0 D0 c0 3

l0 kT

:

ð5bÞ

3. Numerical results The calculated pore morphologies for different treatment times and different ratios dGB/ds are shown in Fig. 2. The following values of dimensionless parameters were employed in calculations: MA = 5, ds = 1, Gs = 1, and GGB = 0.3, where indexes ‘‘GB’’ and ‘‘s’’ in the dimensionless variables refer to the GB and to the surface, respectively, and MA = lA/l0. At a certain treatment time, the leading edge of the pore reaches the surface which results in detachment of the middle grain from the polycrystal. Then the process can repeat itself with the next layer of grains, so that steady-state rate of material removal will be achieved. It can be seen from Fig. 2 that with increasing relative GB diffusivity the pores become more elongated, and the rate of their growth increases. The dependencies of the distance between the leading edge of the pore and the triple junction, l, on the treatment time for different values of dGB are shown in Fig. 3. It is interesting to note that for about half of their lifetime the pores grow with nearly constant velocity. This is counterintuitive since the decreasing distance between the leading pore edge and the surface source of vacancies means, according to Eq. (1), increasing GB flux of vacancies and growth velocity. Yet the pore morphologies shown in Fig. 2 clearly demonstrate the ‘‘sharpening’’ of the edge as it approaches the free surface, which leads to the increase of lB, which in turn partly compensates for the decrease of lAB. At very short distances of the leading pore edge from the surface the increase in gradient of vacancy concentration prevails, and the velocity of pore growth diverges. In our previous work we presented numerical results demonstrating that the lifetime of the middle grain decreases with increasing dimensionless vacancy supersaturation at the surface [7]. The goal of the present work was to find an analytic functional dependence of the lifetime of the middle grain (and, hence, of the rate of steady-state dissolution of the nanocrystalline material) on all relevant parameters of the system. To this end, we derived approximate relationships for the pore front velocity fitting the results of numerical simulations for the initial and final stages of pore growth.

L. Klinger et al. / Computational Materials Science 76 (2013) 37–42

39

Fig. 2. The morphologies of the triple junction pore for the following dimensionless GB diffusion coefficients of vacancies, dGB, and treatment times s (larger pores correspond to longer times): (a) dGB = 0.5, s = 0, 1200, 3600, 5800, 6560; (b) dGB = 1, s = 0, 600, 1500, 2350, 2560; (c) dGB = 2, s = 0, 400, 450, 900, 990; (d) dGB = 3, s = 0, 240, 400, 500, 565.

ðL  lAB Þ2 ðL þ 2lAB Þ ¼ 3At;

ð7aÞ

with

A¼2

mGB DGB XlA : akT

ð7bÞ

This dependence is shown in Fig. 4. In the lengths range 0.3L < (L  lAB)/L < 0.7L it is close to linear. Of course, for smaller lAB Eq. (7a) is not valid, but we can roughly estimate the upper limit of time needed for the pore to arrive at the specimen surface (i.e. the lifetime of the middle grain in Fig. 1), tup, as

sup 

Bt up L4

¼

B a ¼ ðdb mA Þ1 ; 3LA 6

ð8Þ

mGB DGB : ms Ds

ð9Þ

where Fig. 3. The dependencies of the distance between the leading edge of the pore and the triple junction O, l, on the treatment time, s, for different values of dGB.

mA ¼

lA Xcs

L; db ¼

4. Analytical models of pore growth 4.1. Short treatment times For short treatment times the overall size of the pore is so small that all vacancies arriving from the surface along the GB AB (see Fig. 1) are swiftly re-distributed along the internal surface of the pore a(L  lAB), where L is the total length of the near-surface GB (AO in Fig. 1) and a is a geometrical factor varying from 2 to 3 depending on the pore shape. The velocity of pore growth, V, is then

V aðL  lAB Þ ¼ jGB X;

ð6Þ

where jGB is given by Eq. (1). Our numerical results demonstrate that during the initial stages of pore growth lA  lB and, hence, the contribution of the latter can be neglected in our approximate analysis. Taking into account that V = dlAB/dt yields the following relationship for lAB

Fig. 4. The dependence of the relative distance between the leading edge of the pore and the triple junction on the treatment time, calculated employing Eq. (7a).

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L. Klinger et al. / Computational Materials Science 76 (2013) 37–42

For a proper comparison of the lifetime estimates for the short and long treatment times we introduced here Mullins’ coefficient, B, defined according to [17]

B¼

ms Ds X2 cs kT

ð10Þ

:

4.2. Long treatment times For the long annealing times (i.e. when the distance between the pore front and the free surface is much smaller than the grain size), the front of the growing pore is very sharp (see Fig. 2), and most of the vacancies coming from the GB are redistributed in the vicinity of this front. In the steady-state regime this leads to formation of a long and narrow crack-like slit at the GB with the width of h  ðB=jGB XÞ1=2 [18,19]. In this regime, most of the vacancies arriving from the GB are absorbed by the advancing slit-like pore, and diffusion along the long and flat pore walls does not play any role. The chemical potential at the front of the growing pore is then lB  Xcs =h, and thus the GB vacancy flux can be found from Eq. (1)

  mGB DGB XlA Xc ¼ 1  s ðjgb X=BÞ1=2 : kTlAB lA

jGB

ð11Þ

This equation can be solved with respect to jGB

 1=2 jGB X 2mA =L pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ B 1 þ 1 þ 4mA lAB =Ldb

ð12Þ

With this value of vacancy flux the pore growth velocity can be estimated as



dlAB j X ðj XÞ3=2 ¼ V ¼ GB ¼ GB 1=2 dt h B B

¼

L3

2mA pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ 4mA lAB =Ldb

!3 :

Introducing a new dimensionless u ¼ 4mA lAB =db L ¼ 4lA lAB =Xcs db yields

Z

pffiffiffiffiffiffiffiffiffiffiffi 3 1 þ xÞ dx ¼ 32m4A ðslf  sÞ=dgb ;

u

ð1 þ

0

ð13Þ variable

ð14Þ

Here slf is unknown dimensionless lifetime of the pore, since the integration in Eq. (14) is performed from lAB = 0 (i.e. for the pore which reached the surface). In other words, slf  s is the time left before detachment of the middle grain. The dependence of this time on parameter u (which convolutes the surface supersaturation of vacancies, and GB and surface diffusivities) is shown in Fig. 5. Extrapolating the pore kinetics described in this Section to the whole range of treatment times yields the lower estimate of the lifetime of the middle grain, slow. This time can be calculated employing Eq. (13)

slow ¼

Z 0

1

1þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 1 þ 4mA x=db dx  0:4ðdb mA Þ3=2 : 2mA

Fig. 5. The dependence of dimensionless time before detachment of the middle grain on parameter u, calculated according to Eq. (14).

time for dGB = 1, calculated numerically employing the model of Section 2, together with the predictions of the analytical models for the short and long treatment times. A good matching between the results of analytical models and results of numerical calculations in the respective time ranges can be seen. We found that the results of numerical simulations for the lifetime of the middle grain are closer to the lower limit estimate (see Eq. (15)) than to the upper one (Eq. (8)). This is understandable since the upper lifetime limit was estimated based on the model which does not take into account the sharp acceleration of the pore growth rate as it approaches the free surface (see Fig. 6). Therefore, in our further analysis we will employ the lower lifetime limit estimate given by Eq. (15). In our previous work we demonstrated that for the reasonable surface vacancy supersaturations, the proposed mechanism of near-surface porosity predicts the dissolution rate of about 4 nm/s of nanocrystalline Cu with the grain size of 25 nm at room temperature. In the present work we will present a more accurate estimate based on Eq. (15) for nanocrystalline Fe. Iron and ironbased alloys are increasingly considered as materials for bioresorbable implants such as stents and bone scaffolds [20]. As in our previous work [7] we will assume that the maximum vacancy concentration in the material corresponds to the equilibrium vacancy concentration at its melting point. With the vacancy

ð15Þ

Finally, combining Eqs. (8) and (15) yields the following result for the lifetime of the middle grain

0:4ðdb mA Þ3=2 < slf <

a 6

ðdb mA Þ1 :

ð16Þ

5. Discussion and conclusions Fig. 6 demonstrates the dependence of the distance between the leading edge of the pore and the triple junction, l, on the treatment

Fig. 6. The dependencies of the distance between the leading edge of the pore and the triple junction O, l, on the treatment time, s, for dGB = 1 (solid line), together with the results of analytical models for the short (Eq. (7a)) and long (Eq. (14)) treatment times (dashed lines). All other relevant parameters are the same as in Figs. 2 and 3.

L. Klinger et al. / Computational Materials Science 76 (2013) 37–42

formation energy of 1.72 eV [21] and surface energy cs = 1.75 J/m2 [22] in the bcc Fe, and L = 50 nm, this yields mA  470 at room temperature (see Eq. (9)). With the Arrhenius parameters for self-diffusion of Fe adatoms on the surface of Fe whiskers obtained at the temperatures below 250 °C in ultrahigh vacuum, the surface selfdiffusion coefficient of Fe at room temperature is 1.75  1015 m2/s [23]. This results in B = 2.2  1033 m4/s for Mullins coefficient (see Eq. (10)), assuming that all surface atoms are mobile. The largest uncertainty in our estimates is related to the relative GB diffusivity db (see Eq. (9)). Assuming the vacancy diffusion mechanism along the GBs, and the equilibrium vacancy concentration at the room temperature, the product mGBDGB can be estimated by m0b Db , where m0b is the number of atoms per unit area of the GB, and Db is the GB self-diffusion coefficient of atoms. Assuming m0b  ms and estimating Db from the recent data of Inoue et al. [24] obtained on the coarse grain, equilibrated polycrystalline Fe yields very long lifetimes for the middle grain, which means that nanocrystalline Fe is stable against disintegration by grain detachment. However, it is unlikely that the diffusivity of GBs in nanocrystalline Fe in the close vicinity of a very potent surface source of vacancies will be identical to the GB diffusivity in the equilibrated coarse grain polycrystal. For example, it is known that some GBs in ultrafine grain metals and alloys processed by severe plastic deformation exhibit anomalously high diffusivities well above those in coarse grain polycrystals [25–27]. These high diffusivities were attributed to the excess free volume associated with these non-equilibrium GBs [26]. Assuming db  1 as an upper limit for the GB diffusivity yields tlow  0.1 s for the lifetime of the middle grain, which is equivalent to the dissolution rate of 250 nm/s. Thus, dissolution by grain detachment is a viable mechanism of room temperature dissolution of nanocrystalline Fe driven by surface vacancy supersaturation and pores growth along the GBs. It should be noted that surface diffusion in an electrolyte during anodic dissolution may be even faster than in ultrahigh vacuum due to the presence of the electrical double layer (‘‘electrochemical annealing’’ effect [28]). This will further increase the estimated dissolution rate. The dissolution mechanism discussed in the present work is different from the classical intergranular corrosion process which also leads to grain detachment, or ‘‘chunk effect’’ [29,30]. The preferential corrosion attack at the GBs during intergranular corrosion is associated with the higher dissolution rate of the near-GB region as compared to the grain interior [31,32], and grain detachment occurs due to the growth of narrow GB grooves from the external surface toward material interior. On the contrary, the mechanism discussed in the present work describes the growth of elongated GB pores from the nucleation site inside the sample toward its external surface. This mechanism can be recognized by the presence of isolated, elongated GB pores on the cross-sectional micrographs of the near-surface region of corroded sample. Such isolated pores are present in the micrographs of corroded samples of stainless and low-alloy steels published in the literature [32,33]. The dissolution mechanism considered in the present work is relatively slow because it is controlled by diffusion in the solid state. For example, it follows from Eq. (15) that the lifetime of a grain depends on the grain size, L, according to the L5/2 law. This means, for instance, that increasing the average grain size of nanocrystalline iron sample from 50 nm to 500 nm will decrease the dissolution rate from 250 nm/s down to 8 nm/s. Yet we believe that this mechanism is especially relevant for the bulk nanocrystalline materials produced by severe plastic deformation [26]. In these materials, the GBs newly formed during plastic deformation are free from impurities [25] and, therefore, are not prone to preferential galvanic corrosion. Moreover, the high density of defects in the interior of the grains in as-processed material accelerates their dissolution, thus leveling off the difference in dissolution rates

41

between the GBs and grain interior. These factors reduce the rate of classical intergranular corrosion and increase the uniformity of the dissolution front, as has been observed in several studies [34,35]. At the same time, the intrinsic nanometer-scale pores present in nanocrystalline materials produced by severe plastic deformation [26] can serve as potent nucleation sites for elongated GB pores developing beneath the dissolving surface and eventually leading to dissolution by grain detachment. In conclusion, in this work we demonstrated that excess vacancies produced on the surface of a polycrystalline specimen (by oxidation or anodic dissolution) may cause GB cavitation and pores growth in the subsurface regions. We demonstrated that the pores growing from the subsurface triple junction of the GBs toward the surface source of vacancies become more narrow and elongated with increasing GB diffusivity of vacancies. Once the pore reaches the free surface of the specimen the whole near-surface grain detaches from the polycrystal, causing a dramatic increase of the polycrystal dissolution rate. We proposed two simplified analytical models describing the initial and final (just before the grain detachment) stages of pore growth, and estimated the lifetime range of the detaching grain. Our estimates demonstrated that the proposed mechanism can operate in nanocrystalline iron at room temperature. Acknowledgments The author gratefully acknowledge the financial support by the European Commission’s 7th Framework Programme under the project ‘‘Virtual Nanotanium-VINAT’’ (Contract No. 295322), coordinated with the State Contract No. 16.523.12.3002 of the Russian Ministry of Education and Science. The authors are grateful to Prof. E.Y. Gutmanas, Department of Materials Science and Engineering, Technion, for fruitful discussions. References [1] S. Perusin, B. Viguier, D. Monceau, L. Ressier, E. Andrieu, Acta Mater. 52 (2004) 5375–5380. [2] D. Oquab, N. Xu, D. Monceau, D.J. Young, Corros. Sci. 52 (2010) 255–262. [3] H. de Monestrol, L. Schmirgeld-Mignot, S. Poissonnet, C. Lebourgeois, G. Martin, Interf. Sci. 11 (2003) 379–390. [4] R.W. Revie, H.H. Uhlig, Acta Metall. 22 (1974) 619–627. [5] D.A. Jones, A.F. Jankowski, G.A. Davidson, Mater. Trans. 28A (1997) 843–850. [6] A.M. Gusak, Diffusion-Controlled Solid State Reactions, Wiley-VCH Verlag, Weinheim, 2010. [7] L. Klinger, I. Gotman, E. Rabkin, Scripta Mater. 67 (2012) 352–355. [8] P.L. Liu, J.K. Shang, Scripta Mater. 53 (2005) 631–634. [9] L. Klinger, E. Rabkin, J. Mater. Sci. 46 (2011) 4343–4348. [10] L. Klinger, E. Rabkin, Acta Mater. 59 (2011) 1389–1399. [11] A. Nel, T. Xia, L. Madler, N. Li, Science 311 (2006) 622–627. [12] B.S. Zolnik, A. González-Fernández, N. Sadrieh, M.A. Dobrovolskaia, Endocrinology 151 (2010) 458–465. [13] A.M. Schrand, M.F. Rahman, S.M. Hussain, J.J. Schlager, D.A. Smith, A.F. Syed, WIREs Nanomed. Nanobiotechnol. 2 (2010) 544–568. [14] V.E. Kagan, H. Bayir, A.A. Shvedova, Nanomed.: Nanotechnol. Biol., Med. 1 (2005) 313–316. [15] R.M. Urban, J.J. Jacobs, M.J. Tomlinson, J. Gavrilovic, J. Black, M. Peoc’h, J. Bone Joint Surg. (Am) 82A (2000) 457–477. [16] E. Dornel, J.-C. Barbé, F. de Crécy, G. Lacolle, J. Eymery, Phys. Rev. B 73 (2006) 115427. [17] W.W. Mullins, J. Appl. Phys. 30 (1959) 77–83. [18] T.-J. Chuang, J.R. Rice, Acta Metal. 21 (1973) 1625–1628. [19] L.M. Klinger, E.E. Glickman, V.E. Fradkov, W.W. Mullins, C.L. Bauer, J. Appl. Phys. 78 (1995) 3833–3838. [20] M. Moravej, A. Purnama, M. Fiset, J. Couet, D. Mantovani, Acta Biomater. 6 (2010) 1843–1851. [21] M.I. Mendelev, Y. Mishin, Phys. Rev. B 80 (2009) 144111. [22] J.W. Martin, R.D. Doherty, B. Cantor, Stability of Microstructure in Metallic Systems, second ed., Cambridge University Press, 1997. [23] J.A. Stroscio, D.T. Pierce, Phys. Rev. B 49 (1994) 8522–8526. [24] A. Inoue, H. Nitta, Y. Iijima, Acta Mater. 55 (2007) 5910–5916. [25] Y. Amouyal, S.V. Divinski, Y. Estrin, E. Rabkin, Acta Mater. 55 (2007) 5968– 5979. [26] S.V. Divinski, G. Wilde, E. Rabkin, Y. Estrin, Adv. Eng. Mater. 12 (2010) 779– 785.

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