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Acta Materialia 56 (2008) 165–176

Williamson–Hall anisotropy in nanocrystalline metals:X-ray diffraction experiments and atomistic simulations

S. Brandstetter, P.M. Derlet, S. Van Petegem, H. Van Swygenhoven *

Paul Scherrer Institut, ASQ/NUM – Materials Science & Simulation, PSI-Villigen CH-5232, Switzerland

Received 30 April 2007; received in revised form 24 July 2007; accepted 6 September 2007Available online 23 October 2007

Abstract

X-ray diffraction techniques allow the determination of mean grain size, root-mean-square strain and dislocation/twin contentthrough peak profile analysis and the Williamson–Hall approach. The analysis is, however, based on assumptions that are questionablewhen applied to nanocrystalline (nc) structures. In the present work, two-theta X-ray diffraction profiles are calculated from multi-mil-lion atom computer-generated nc-Al samples containing a well-defined twin or dislocation content. The Williamson–Hall plots are com-pared with those obtained from in situ X-ray experiments performed on electrodeposited nc-Ni and on nc-Cu data available in theliterature.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Nanocrystalline materials; Molecular dynamics; X-ray diffraction (XRD)

1. Introduction

X-ray diffraction (XRD) peak profile analysis is oftenused to characterize the microstructure of nanocrystalline(nc) metals [1–3], in particular to determine the mean grainsize and the root-mean-square (rms) strain [4,5]. William-son and Hall [6] demonstrated that the integral width(IW) of a 2h diffraction peak can be decomposed into acontribution due to a finite scattering volume and a contri-bution due to local fluctuations in strain. The IW is for-mally defined as the integral of the peak profile dividedby the peak height. In terms of the scattering vector,s = 2sinh/k, the IW can be written as

ds ¼ dsSize þ dsStrain; ð1Þwhere ds = 2coshdh/k and 2dh is the measured IW. HeredsSize = 1/LSize, where LSize is the length that characterizesthe coherent scattering volume, and dsStrain ¼ 5

ffiffiffiffiffiffiffiffihe2i

ps=2.

Hereffiffiffiffiffiffiffiffihe2i

pis the rms strain [4]. Thus, Eq. (1) predicts a lin-

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.09.007

* Corresponding author. Tel.: +41 56 3102931; fax: +41 56 3103131.E-mail addresses: helena.vs@psi.ch, helena.vanswygenhoven@psi.ch

(H. Van Swygenhoven).

ear relation between peak IW and peak order (s = |ghkl|,where ghkl is the reciprocal lattice vector corresponding tothe hk l reflection), and that the y-intercept is equal to theinverse of LSize, which in the present work is taken as thecharacteristic grain size. Eq. (1) in fact corresponds to aCauchy–Cauchy deconvolution of the peak broadeningdue to grain size and rms strain. In practice, however, aCauchy–Gaussian is more often used [5], resulting in amonotonic non-linear relationship between the three quan-tities in Eq. (1). To take into account the presence of twinfaults, Warren modified the left-hand side of Eq. (1) tods� aW ghkl

, where a is the probability that a 111 plane isa twin fault and W ghkl

is a geometrical factor describingits effect on the IW of the particular h k l diffraction peak[7,8]. A similar modification can also be used to extractthe stacking fault density [8].

In this procedure, only two diffraction peaks of the samecrystallographic family are needed to separate the effect ofgrain size and rms strain on the peak broadening, when thetwin density is neglected or of a known value. The obtainedvalues for grain size and rms strains are, however, typicallydependent on the chosen family, i.e. whether 111/222 or200/400 is used [5], reflecting a non-monotonic increase

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166 S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176

of the IW vs. scattering vector, an effect that is oftenreferred to as the anisotropy in the Williamson–Hall(WH) plot. Such WH anisotropy has been interpreted interms of a lattice dislocation [9,10] and/or twin fault [7,8]content. The introduction of dislocation contrast factors[11] has allowed the development of profile analysis proce-dures that can result in a quantitative determination of thedislocation density, grain size and twin density, a proce-dure called the modified WH analysis. Given a particulardislocation distribution (in terms of spatial distributionand dislocation character), such contrast factors providea measure of how strongly the dislocations can affect theRMS strain within the system and thus the strain-broad-ened component of the Bragg diffraction peak profile.These methods are, in part, based on the earlier works ofKrivoglaz et al. [12] and Wilkens and Bargouth [13]. Suchan approach typically assumes a particular geometry of dis-locations with one Burgers vector magnitude of a particu-lar screw/edge character. In addition to such analyticalprofile fitting methods there exist a wide variety of alterna-tive approaches to extract microstructural informationfrom XRD spectra, ranging from the early Fourier methodof Warren and Averbach [14,15], full profile analysis meth-ods [16], maximum entropy methods [17] and more gener-ally statistical techniques that investigate more closely the‘‘tail’’ structure of the Bragg peaks such as the secondand fourth moment analyses [18,19], which can relate peakshape to a dislocation density, a characteristic density fluc-tuation and dislocation–dislocation correlation length. Wenote, however, that full profile analysis methods are oftennot applicable to in situ experimental data since often onlya limited range of scattering data is available.

In the literature reporting experimental results, the fullwidth at half maximum (FWHM) is often used to representthe width of a diffraction peak. The FWHM exhibitssimilar anisotropic behaviour as the IW, but is generally10–20% smaller and the IW values depend less on the lineprofile used. Fig. 1 displays the WH plot for electrodepos-ited (ED) Ni with a mean grain size of �30 nm [20] and for

Fig. 1. Experimental WH plot of electrodeposited nc-Ni [20] and inert gascondensed nc-Cu [21].

inert gas condensed (IGC) Cu with a mean grain size of�22 nm [21]. An anisotropic progression of the peakwidths with diffraction order is evident, but very differentfor the two types of nc metals. Using Eq. (1), the meangrain size will be clearly dependent on the choice of thereflection family, where for both Ni and Cu the grain sizeobtained using the 111/222 family is larger then the valueobtained using the 200/400 family. Applying the peakprofile analysis to nc-Cu [7,8], a dislocation density of�4.7 · 1015 m�2 is obtained, which corresponds to a graincontaining either zero, one or two dislocations [21]. Thesame work derives a twin density probability a of �0.012giving a mean separation distance between twins of�17 nm. Transmission electron microscopy (TEM) investi-gation of similarly prepared nc-Cu confirmed a consider-able number of twins, where some grains were observedto contain several twins [22]. For the case of ED nc-Ni, asimilar XRD analysis [23] gives an average dislocation den-sity of �4.9 · 1015 m�2 and a twin density probability of0.0012, corresponding with a mean twin separation of�170 nm and a mean grain size of 50 nm. TEM analysisof ED nc-Ni confirms the much lower twin density; how-ever, it does not support the high lattice dislocation contentin grain interiors [2], although some dislocations close tothe grain boundary (GB) region have been identified [24].

Many of the simplifying assumptions used in both thedislocation and twin analysis procedures may, however,no longer be justified due to the small grain size and thecorresponding high interface density. In the case of theWarren twin analysis procedure, one of the essential crite-ria for applicability is that twin fault densities are equal andtherefore statistically meaningful for each grain [8,25]. Theassumption of a particular dislocation signature in terms ofsingle-crystal geometry and Burgers vector content mightnot be correct in an nc grain, where simulations predicthighly curved, finite-length dislocations of spatially varyingcharacter [26].

It has been generally accepted that during plastic defor-mation of nc face-centred cubic (fcc) metals, GBs act asboth a source and sink for dislocations. This mechanismhas been suggested by molecular dynamics (MD) computersimulations [26–29], where dislocations are emitted fromGBs, travel through the entire grain, and are eventuallyabsorbed in neighbouring and opposite GBs. The existenceof such a mechanism is experimentally supported at leastfor ED Ni by the lack in a TEM-visible dislocation net-work and the observation of a reversible XRD peak broad-ening after unloading of a sample deformed with 6% tensilestrain [20]. Extensive research into the behaviour of peakprofiles during tensile loading and unloading cycles hasdemonstrated that the microplastic regime in a tensile testextends to relatively high total strains, and that a time-dependent recovery of the peak widths sets in after the fullplastic regime has been reached [30]. Such behaviour hasbeen explained by the ability of the GBs to accommodatedislocations. It should be noted that all experiments havebeen performed in tension, where the maximum total strain

Fig. 2. nc-Al strain vs. simulation time showing the initial loading andsubsequent unloading regimes for the creation of samples (DIS-1 to DIS-3). For the loading curve the applied uniaxial tensile stress is 1.6 GPa.

S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176 167

before failure is limited to 6% and therefore the amount ofdislocation activity limited.

To contribute to the basic understanding of the charac-ter of the WH anisotropy in nc metals, a method [31] tocalculate the XRD profile from multi-million atom com-puter-simulated samples has been developed. It was shownthat for Voronoi nc networks with generally high-angleGBs and defect-free grain interiors, the calculated XRDpatterns exhibit an almost linear WH plot when the meangrain size is above 10 nm, which is not what is experimen-tally observed. It was therefore concluded that thenon-monotonic behaviour in the experimental samplesmust come from structural properties that have not beenincorporated in the as-prepared simulated structures.

In the present work, the same method is used to investi-gate other possible sources of the anisotropy, by calculat-ing XRD profiles of computer generated nc structureswith a predefined content of lattice dislocations or twinboundaries. It is shown that twins and dislocations exhibita different type of WH anisotropy, and that in an nc GBnetwork lattice dislocations and extrinsic GB dislocationsresult in a similar anisotropic shape of the WH plot. Withthis information in mind, the WH anisotropy of as-pre-pared and deformed ED Ni is investigated. In situ XRDprofile analysis was performed in the Swiss Light Sourceduring compression of the samples to a total strain of25%. The WH plots are compared with measurements onas-prepared and tensile deformed samples. Comparisonsbetween simulations and experiments explain the experi-mentally observed anisotropies and underline the impor-tance of the dislocation mechanism in ED Ni.

2. Atomistic simulations

XRD patterns are calculated from two series ofcomputer-generated nc samples. In the first series, DIS-0,1,2,3, each sample contains a progressively larger disloca-tion content, and in the second series, TWIN-0,1,2,3, eachsample contains a progressively larger twin content. Bothseries of samples contain a fully three-dimensional GB net-work obtained using a Voronoi construction [32,33] andsimulations are performed for Al using the empirical poten-tial of Mishin et al. [34]. Al, rather than Ni or Cu, was cho-sen because the empirical potential allows the nucleation ofboth leading and trailing partial dislocations at the GBduring tensile deformation due to the ratio of the unstableto stable stacking faults being close to unity [28]. Thisresults in the generation of full dislocations within the ncgrains. Note that Al is less elastically anisotropic than Niand Cu, which will be shown to lead to some minor quan-titative differences in terms of rms strain.

2.1. Computational samples

2.1.1. Dislocation samples

The samples used for the investigation of lattice disloca-tion content are all derived from one nc-Al configuration,

DIS-0, that contains 100 defect-free grains (in total �5 mil-lion atoms) with a mean grain size of 10 nm. The 10 nmgrain size was chosen for computational reasons and alsoso that the results could be directly compared to past sim-ulation work [26–28]. This sample was constructed via theVoronoi method as described in Refs. [32,33] and then fur-ther relaxed using 100 ps of molecular dynamics at 800 Kfollowed by 20 ps of equilibration at 300 K. This final sam-ple is referred to as DIS-0. By applying a constant uniaxialtensile stress of 1.6 GPa to this sample, an initial strain rateof �7 · 108/s was obtained during the first 60 ps, increasingto a value of �1.7 · 109/s at 80 ps of deformation as shownin Fig. 2. Such a rapid increase in strain rate beyond 80 psis related to the activation of different slip systems withinone grain, together with the observation of some mechan-ical twinning close to triple junction regions by means ofsequential partial dislocation activity from GBs [35]. Wenote that the activation of multiple slip systems and theobservation of different slip mechanisms originate fromthe high strain rates of simulations [28], in which othermechanisms outside the timescale of the simulation areunable to accommodate the plastic strain. At the largeststrain in Fig. 2, a total of 180 full lattice dislocations wereobserved. We have previously always studied configura-tions that were deformed at the lower strain rates, resultingin mostly the activation of one slip system per grain [26].Here, the goal was to obtain samples with a considerabledislocation content, justifying the use of the higher strainrates. Such configurations are unloaded by instantaneouslyreducing the applied load to zero and performing roomtemperature MDs for 20 ps. Fig. 2 also displays the corre-sponding unloading curves starting from three differentplastic strains, resulting in the unloaded configurationsDIS-1, DIS-2 and DIS-3.

Table 1 presents a detailed atomic scale investigation ofthe dislocation structures of the three unloaded samples interms of the total number of remaining full and partial dis-locations, and their proximity to the GB. It is important to

Table 1Dislocation number and type for unloaded samples using classification inFig. 3

Sample DIS-1 DIS-2 DIS-3

Number of dislocations 39 60 70

Number of full dislocations 18 36 42% in grain interior 78 58 52% close to grain boundary 22 42 48

Number of partial dislocations 31 24 28% clearly visible 81 79 79% where core segment is only visible 19 21 21

168 S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176

note that in spite of the short unloading times, a large per-centage of lattice dislocations are absorbed in the GB: inDIS-3, 110 of the 180 lattice dislocations present beforeunloading are absorbed in the GBs. The remaining disloca-tions are categorized into four classes, as displayed inFig. 3. In Fig. 3, atoms are coloured according to a med-ium-range order analysis [36] where grey represents atomsthat are locally fcc, red1 atoms that are locally hexagonalclose packed (hcp), green other 12 coordinated atoms,and blue non-12 coordinated atoms. Fig. 3a and b displaystwo classes of full lattice dislocations: Fig. 3a shows dislo-cations well within the grain interior, and Fig. 3b shows fulldislocations close to the GB. Fig. 3c and d represents bothpartial dislocations subclassified as: clearly visible partialdislocations close to the GB (Fig. 3c); and the left-over coresegments of a dislocation that is not fully absorbed(Fig. 3d). Note that the class represented in Fig. 3d canbe identified with either nucleation of a partial or absorp-tion of the trailing partial; in both cases a ‘‘lattice Burger’svector’’ content can be identified. Finally, note that the per-centage of multiple slip in grains increases in DIS-1, DIS-2and DIS-3 (10%, 15% and 17%, respectively).

2.1.2. Twin samples

The nc-Al twin samples consist of 15 fcc grains (in total�1.2 million atoms) with a mean grain size of 12 nm inwhich twins transecting entire grains are geometricallyintroduced as described in Ref. [37]. The four samples con-sidered have precisely the same grain shape and GB struc-ture but different twin content: TWIN-0 contains no twins,in TWIN-1 7 of the 15 grains contain one twin, and inTWIN-2 all 15 grains contain one twin. Finally, TWIN-3has a twin content defined by a twin probability, a, definedin the Warren twin analysis theory [7,8] where for one par-ticular 11 1 direction, each 111 plane has an uncorrelatedprobability a of being a twin, leading to a mean separation1/a between the twins. For TWIN-3, a value of a = 0.1 wasused, resulting in a microstructure where each grain maycontain several twins. Relaxation and equilibration of the

1 For interpretation of the references to colour in Figs. 3, 4, 8 and 11, thereader is referred to the web version of this article.

geometrically constructed samples was done as in Refs.[32,33]. In Fig. 4b, a picture of the TWIN-3 sample is givenwhere (red) hcp 1 11 planes can be seen transecting the(grey) fcc grains.

2.2. X-ray calculation method

The XRD two-theta spectrum is calculated for the dislo-cation samples DIS-0 to DIS-3 and the twin samplesTWIN-0 to TWIN-3 using the method presented in Ref.[31]. Here, for an incident wavelength, k, the two-thetascattering profile is calculated via:

IðkÞ ¼Z rc

0

dr lðrÞ sin krkrþ 4pq

rc cos krc

k2� sin krc

k3

� �

�Z rc

0

dr lðrÞ sin krkrþ 4pq

rc cos krc

k2ð2Þ

where k is the scattering vector magnitude defined ask = 2psinh/k, l(r) is the interatomic pair correlation func-tion, rc the continuum cut-off distance (which is typicallyslightly larger than the grain size), and q is the bulk density.The final approximation is valid when rc >> k, which willgenerally be the case for the grain sizes considered here.Note that this method determines the XRD spectrum ofa sample that is spherically averaged and therefore cannotbe used to calculate the XRD spectra of a sample underload where the spherical symmetry has been broken.

The calculated two-theta spectra span a range of scat-tering angles that include the 111, 20 0, 220, 311, 222and 4 00 Bragg diffraction peaks for Al. Fig. 4 displaysa close-up of the calculated 111 and 200 peaks forrespectively (a) DIS-0 and DIS-3 and (b) TWIN-0 andTWIN-3, showing that the diffraction peaks broaden withthe introduction of lattice dislocations or twin planes. Atypical full spectrum may be seen in Ref. [31]. To extractintegral peak width information, each peak is fitted to aPearsonVII function. Close inspection of Fig. 4. revealsan oscillatory behaviour at and around each diffractionpeak, which is a result of the relatively narrow and dis-crete grain size distribution contained within the currentsamples-sharp changes in misorientation in real spaceintroduce long-range oscillatory behaviour in reciprocalspace, the effect of which can be averaged out when atruly continuous grain size distribution is present. Despitethis deficiency, each peak could be fitted to a PearsonVIIfunction with a relative error of at most 10% in the peakwidths. Fig. 5 displays the calculated 220 peak profilesalong with their corresponding PearsonVII for samplesDIS-0 and DIS-3. A relatively good fit can be seen,and the residuals reveal the weak oscillatory behavior.Interestingly, the oscillatory behaviour is somewhatreduced for the dislocated sample. Due to the same grainshape and misorientation distribution of the comparedsamples, such errors were systematic, making it possibleto identify trends over different dislocation and twindensities.

Fig. 3. Different classes of dislocation structures in the unloaded samples: (a) a full dislocation well within the grain interior, (b) a full dislocation close to agrain boundary, (c) a partial dislocation attached via its stacking fault to the grain boundary, and (d) a partial core segment in the GB.

Fig. 4. Calculated X-ray diffraction peaks of the 200 and 111 Bragg peaks showing the increase in integral width due to (a) the introduction of latticedislocations and (b) the introduction of twin boundaries.

S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176 169

Fig. 5. Comparison between calculated 220 diffraction peak and analyt-ical peak profile fit, where at the bottom the residuals are displayed using alarger vertical scale resolution.

170 S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176

2.3. XRD of dislocation samples

Fig. 6a displays the peak widths for the 111, 200, 220,311, 22 2 and 400 Bragg peaks as a function of peak posi-tion in the usual WH format in terms of the scattering vectors = 2sinh/k for samples DIS-0 to DIS-3. DIS-0 produces aWH plot that exhibits little anisotropy, confirming the weakstrain fields of the intrinsic GB dislocations within the as-constructed dense nc GB network [31]. For the deformedsamples DIS-1 to DIS-3, an increase in anisotropy isobserved with a pronounced increase in the IW of the(311) peak as the dislocation content increases. Also shownin Fig. 6a is the WH data of the sample prior to the 800 Krelaxation procedure. It demonstrates that the computa-tional annealing procedure, which results in the formationof better-defined GBs and extrinsic dislocations in generalhigh-angle GBs [38], also results in an increase of the WHanisotropy similar to the type obtained from dislocations.

Table 2 displays the mean grain size and rms strain ofthe 111/2 22 and 2 00/400 family of peaks in Fig. 6a usinga Cauchy–Gaussian deconvolution of the correspondingIW. For both families of peaks the calculated mean grain

Fig. 6. The calculated Williamson–Hall plot as a function of incr

size remains approximately constant, whereas the rmsstrain increases roughly linearly with increasing lattice dis-location content. For the 200/40 0 family, the fluctuationsin grain size and the increase in RMS strain as a function ofdislocation content are slightly greater. Due to the shortMD simulation time, changes in the mean grain size areexpected to be negligible. It should be noted that simula-tions involving longer periods of time and larger plasticstrains do occasionally result in localized grain growth[39]. Therefore, the ‘‘rule of thumb’’ method of tradition-ally using the 111 and 222 peaks to derive the XRD esti-mate for mean grain size is validated in the present work.The values of rms strain extracted for DIS-3 are high,but not unreasonably so: similar values for, respectively,(11 1)/(222) and (200)/(40 0) are experimentally obtainedfor ED nc-Ni with a mean grain size of 30 nm [20]. Itshould be noted, however, that much lower values areobtained for IGC-Cu, and that ED nc-Ni differs in othermicrostructural details, such as incorporated impurities.

To ascertain the contribution of lattice dislocations tothe increase in anisotropy seen in Fig. 6a, sample DIS-2was separated into its constituent grains and an XRD spec-trum of each grain was calculated. An average XRD spec-trum was constructed from the XRD spectra of thosegrains that do not belong to the classification containedin Table 1, i.e. those grains containing no observable latticedislocations. Fig. 7 displays the resulting WH plot com-pared to the full spectra of DIS-2 and DIS-0 and demon-strates that the grains that contain no observabledislocations produce a similar WH anisotropy as that seenfor the entire sample DIS-2. In other words, the increase inanisotropy seen in DIS-2 compared to DIS-1 arises not justfrom those grains containing lattice dislocations. Closerinspection of Fig. 7 reveals that the IW of DIS-2 withoutdislocations shifted to higher values than those of the entireDIS-2 sample. This shift arises from the algorithm used toextract the individual grains, which employs only fcc atomswith no nearest neighbours close to the GB, therebyslightly reducing the effective size of the coherent scatteringvolume of each grain.

easing (a) dislocation content and (b) twin boundary content.

Table 2Mean grain size and rms strain derived from WH plot using only the 111/222 and 200/400 family of peaks

Sample 111/222grain size(nm)

111/222 rmsstrain (%)

200/400grain size(nm)

200/400 rmsstrain (%)

DIS-0 10.4 0.16 10.9 0.14DIS-1 10.3 0.27 11.2 0.24DIS-2 10.2 0.34 11.5 0.57DIS-3 10.6 0.42 12.3 0.67

Fig. 7. Williamson–Hall plots of sample DIS-0 and DIS-2. Also shown isthe corresponding Williamson–Hall plot of the average XRD spectrum ofthose grains containing no easily observable dislocation content.

S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176 171

2.4. XRD twinned samples

Fig. 6b displays the peak widths as a function of scatter-ing vector for the series of twinned samples. The introduc-tion of twins increases only slightly the anisotropy when nomore than one twin is present per grain (TWIN-1 andTWIN-2), but the curve shifts to higher peak widths,reflecting the generally smaller coherent scattering volumesassociated with the presence of the twin planes. For TWIN-3, however, characterized by the Warren twin probabilitya = 0.1, a considerably larger anisotropy emerges predom-inantly from a large increase in integral width of the 200and 400 peaks. From Warren’s twin analysis [7,8] it isthe 100 family of peaks that is most affected by the pres-ence of twins and it is the 111 family that is affected theleast. Qualitatively this is due to the fact that for the 111family there is always one 111 direction that is not affectedby the twins, whereas all 100 planes have the same geomet-rical relationship with the 111 plane. That samples TWIN-1 and TWIN-2 do not follow this trend can be understoodqualitatively from the realization that the probabilisticWarren analysis cannot be applied to samples TWIN-1and TWIN-2, since such an analysis considers the peakindex-dependent coherent scattering volume length scaleas the average distance between neighbouring twin planes:TWIN-1 and TWIN-2 contain at most one twin per grainand therefore the coherent scattering volume length scale

is defined by the average distance between the twin andneighbouring GB.

2.5. Internal stress

Atomistic simulations allow the calculation of the inter-nal stress distribution in the sample. Since stress variationswithin the fcc lattice are linearly related to strain varia-tions, such a local stress analysis gives us information onthe origin of the rms strains derived from the previousWH plots.

To investigate the internal stress of the computer-gener-ated samples, the time-averaged local atomic stress of avolume element X is calculated via:

rXlm ¼

1

X

Xi2X

mvi;lvi;m þ1

2

Xi2X;j

F ðrijÞrij;lrij;m

rijlij

* +: ð3Þ

Eq. (3) was developed by Cormier et al. [40] and representsan improvement over the usual virial expression for localstress in that it rigorously satisfies conservation of linearmomentum for the chosen volume element. In the aboveequation, F ðrijÞ ¼ F j~ri �~rjj

� �represents the force magni-

tude between atoms i and j at positions~ri and~rj with cor-responding velocities ~vi and ~vj. Thus, for each chosenvolume element, the stress is determined by the individualatomic virial stress contributions only within the volumeelement (with lij representing the fraction of the bond be-tween atoms i and j within the volume X). The h� � �i repre-sents a time average taken during a MD simulation. Theabove approach has been used in our previous work inwhich the volume element is chosen to be a sphere centredon each atom with a radius that is approximately halfwaybetween the first and second nearest neighbour shell[26,41].

Fig. 8 displays a section of atoms showing a centralgrain and its neighbours. The left column (Fig. 8a,c ande) is the configuration for sample DIS-0 and the right col-umn (Fig. 8b,d and f) is the same configuration for sampleDIS-2. In Fig. 8 the atoms are coloured according to theirlocal crystallinity (Fig. 8a and b), their local hydrostaticpressure (Fig. 8c and d) where blue represents �1.5 GPaor lower and red 1.5 GPa or higher, and their local devia-toric stress (Fig. 8e and f) where blue represents a value of0 GPa and red 1.5 GPa or higher. For intermediate valuesthe linear colour bars shown in Fig. 8 can be used. Notethat a negative value of hydrostatic pressure represents ten-sion and a positive value compression. The deviatoric stresscan be considered as the magnitude of the local maximumresolved shear stress. In sample DIS-2, the central graincontains a full dislocation (circled in yellow) of the typethat is close to the GB (see Fig. 3b). The core structureof the dislocation is clearly visible in Fig. 8b by means ofthe red stacking fault connecting the leading and the trail-ing partials. The viewing direction of the atomic sectionshown in Fig. 8 is along the (111) slip plane (indicatedby the yellow lines intersecting the yellow circle) and nearly

Fig. 8. The (a) local crystallinity, (c) local hydrostatic pressure and (e) local maximum shear of a section of the sample prior to loading (DIS-0). (b), (d)and (f) are the same sections for the loaded/unloaded sample DIS-2. In (b), a full dislocation is circled yellow where the two yellow lines represent the 111slip plane. The yellow arrowed regions indicate stress concentrations that result from the absorption of a lattice dislocation in the GB.

172 S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176

perpendicular to the (110) slip vector of the dislocation.Detailed analysis has shown that this particular full dislo-cation has a predominant edge character with a Burgersvector directed towards the right-hand side of the figure.The edge character is reflected in the stress distributionaround the dislocation: the hydrostatic pressure (Fig. 8d)shows a compressive region below the slip plane (due tothe extra 11 0 half plane) and a tensile region (blue) abovethe slip plane. The deviatoric shear signature does notresolve the shear lobes ahead and behind the full disloca-tion predicted by linear elasticity [42], i.e. the long-rangestress fields associated with the presence of the full disloca-tion. This may be due to the image stresses occurring fromthe nearby GBs and/or the 250 fs time average which doesnot remove all variations in stress arising from thermalactivity.

Looking at the pictures of DIS-0 (Fig. 8c and e) the GBregions contain local stress anomalies (see yellow arrows inFig. 8c) that upon deformation and subsequent unloadingchange their configuration. For example, comparison of

Fig. 8c and d (DIS-2) shows that some of the displayedGB regions have relaxed to a configuration predominantlyunder a tensile pressure, whereas new regions under com-pressive hydrostatic pressure have emerged.

Since the WH plot of DIS-0 shows nearly no anisotropy,whereas a pronounced anisotropy is observed for DIS-2, itis worth investigating the origin of the newly developedstress intensities during deformation. Fig. 9 displays thesame atomic section as in Fig. 8 but showing the displace-ment vectors between the atom positions of DIS-2 andDIS-0. The colour of the displacement vectors representsthe magnitude of the distance moved: dark blue represents0 and red >5 A. The black arrows indicate slip activity inneighbouring grains resulting in a dislocation being depos-ited in the GB. The stress anomaly regions arrowed inFig. 8d and f are found to correlate well with the resultingdeposited dislocation structures within the GB, whereas thestress intensities indicated in Fig. 8c (DIS-0) are relatedwith GB dislocations and not with slip activity. Stressintensities such as those in Fig. 8d and c are, however,

Fig. 9. Displays the atomic displacement vectors between the twoconfigurations (a) and (b) in Fig. 8, in which the colour of the linesrepresent the displacement distance. The three arrowed regions indicate111 planes within neighbouring grains that have undergone slip due to thepassage and absorption in the grain boundary of a full dislocation.

Fig. 10. Corrected stress vs. strain loading curve under compression fornc-Ni.

Fig. 11. Williamson–Hall plots for experimental ED nc-Ni (a) as-prepared, (b) unloaded from tensile deformation and (c) unloaded fromcompression deformation.

S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176 173

not counted as part of the dislocation content as defined inTable 1. In other words, in addition to the ‘‘easily’’ identi-fiable lattice dislocation content as defined by the four cat-egories in Fig. 3, there are also dislocation signatureswithin the GB region that contribute to the WH anisotropythat are only visible through their stress concentrations.

3. Experimental peak profile analysis: ED Ni

3.1. In situ XRD during compression

Room temperature in situ compression experimentswere performed at the Material Science beam line of theSwiss Light Source. The X-rays of this facility have anenergy of 17.5 keV and are detected by a multistrip detec-tor allowing full 60� diffraction spectra to be collected withan intrinsic angular resolution of 0.004� at a frequency of0.1 Hz. The compression tests were performed on a modi-fied version of the miniaturized tensile machine describedin detail in Ref. [43]; for the current experiment the tensilegrips were replaced by flat compression heads and the sam-ples had a rectangular shape with dimensions of0.6 · 0.6 · 0.2 mm3. The strain was measured using ahigh-resolution digital camera with image recognition soft-ware and corrected for the compliance of the machineusing the Young’s modulus of nickel. All compression testswere performed at a strain rate of 1.6 · 10�4 s�1. A typicalstress–strain curve is shown in Fig. 10. Large strains up to25% could be obtained in contrast to the restricted 6%strain of the tensile experiments.

3.2. WH plot on compressed samples

Fig. 11 shows the WH plot from the as-prepared ED Nisample (black), an unloaded sample that has beendeformed under tensile conditions to 6% total strain [20]

(red) and an unloaded sample that has been uniaxiallycompressed until 25% strain (green). Note that for theexperiment the FWHM is used in order to be able to makecomparisons with literature data. The anisotropy observedin the as-prepared sample is similar to previous work [20]and changes very little during tensile deformation [30],but there is a considerable difference with the anisotropyobserved after compression to a total strain of 25%. Whileupon unloading the peak widths of the 200 and the 222peaks recover slightly to lower values than those prior todeformation, a clear increase in peak widths is observedfor the 311 peak. The lack of 311 recoverability has alsobeen observed in previous in situ tensile experiments [30]and at this point there exists no comprehensive answer towhy this is so. The XRD profiles of the deformed sampleshave been recorded immediately after unloading of thesample. Similar to what was observed during tensile defor-mation, a time-dependent relaxation of the peak broaden-ing for all diffraction peaks as a function of unloading timewas seen. This type of relaxation occurs for all diffractionpeaks and amounts to reductions of only 2–4% in the

Fig. 12. (a) A TEM picture of a specimen that experienced 6% tensile deformation until 6% strain. (b) A TEM picture from the same material, but thistime deformed in compression until 25% strain, is shown on the right-hand side.

174 S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176

values of the peak widths after unloading without changesin the degree of anisotropy.

TEM analysis of both the tensile and compressivedeformed samples did not reveal any major difference inthe microstructures. Fig. 12 shows a representative TEMimage of both samples. Extensive studies of grain size dis-tributions [44] show no measurable changes in the micro-structure and the mean grain size. Moreover, previoushigh-resolution TEM work has demonstrated no statisti-cally meaningful dislocation content before and afterdeformation, and only if the material is deformed at liquidnitrogen temperatures do a reasonable number of disloca-tions become visible [45]. It must, however, be noted thatsuch TEM work can suffer from thin film artefacts due tosample preparation.

4. Discussion

In a computer-generated nc system constructed fromdefect-free grains of random orientation with an averagegrain size of 12 nm, the integral width vs. peak positionexhibits minor anisotropy despite the high interface densityand the local stress intensities observed in the GBs. Byintroducing dislocations or twins, the WH plot developsan anisotropy that is characteristic for both types of defects.

Comparison of the WH anisotropy arising from theintroduction of twins (Fig. 6b) with that of experimentalnc-Cu (Fig. 1) suggests that the closest resemblancebetween experiment and simulation can be found in theheavily twinned computer-generated sample, TWIN-3.Although the experimental twin faulting probability a is 8times smaller then that of TWIN-3, both WH plots exhibitthe expected strong 111/200 anisotropy predicted by theWarren analysis procedure. This can be understood byrecalling that the grain size for the simulated sample is�10 nm, whereas for the experimental nc-Cu sample it is�22 nm, resulting in approximately the same number oftwins per grain. Furthermore, the WH plot shifts to higherabsolute values of the peak widths, reflecting the reduction

in coherent scattering volume in the presence of twinlamellae.

However, the simulations show that when the micro-structure of the sample is dominated by dislocations, theslope of the WH plot increases and develops an anisotropythat is dominated by a broadening of the (311) diffractionpeak, exhibiting the greatest increase in peak width. Thistype of anisotropy is also developed during compressionof the ED Ni to large plastic strains. Inspection of theWH plot after 6% tensile deformation shows the sametrend though to a lesser degree: only very small changesin peak width are observed. When comparing the WH inthe simulated and the experimental sample, one alsonotices, however, a difference. The ratio of the peak widthof the (111) to the (200) peak is much smaller for ED Nithan for computational Al. This can be explained by thedifference in elastic anisotropy between Al and Ni whichaffects the dislocation contrast factors [11].

Careful analysis of the computational samples has dem-onstrated that a ‘‘dislocation’’ type WH anisotropy is notonly obtained when lattice dislocations are present in thegrain interior, but can be also induced by local stress inten-sities in the GB resulting from the absorption of lattice dis-locations. These stress intensities have long-range stressfields and therefore contribute to the rms strain of the sam-ple, whereas the GB dislocations that resulted from samplerelaxation in a Voronoi construction do not contribute tothe rms strain. It is interesting to notice that when the com-putational sample is annealed after Voronoi construction,the WH anisotropy also starts to become visible.

The TEM ‘‘visible’’ lattice dislocation densities (disloca-tions of the type seen in Fig. 3a and b) of DIS-2 amounts to�8 · 1015 m�2 and is somewhat higher than the dislocationdensity calculated for as-prepared ED Ni using profileanalysis procedures employing dislocation contrast factors[23]. The fact, however, that the anisotropy changes verylittle when omitting the grains containing these lattice dis-locations suggests that at these small grain sizes the WHapproach with its particular assumptions for lattice dislo-

S. Brandstetter et al. / Acta Materialia 56 (2008) 165–176 175

cation geometries is not applicable for nc metals. Moreoverthe simulations demonstrate that there is no need for aneasily ‘‘visible’’ lattice dislocation content to explain theexperimentally measured WH anisotropy in as-preparedand deformed samples. Previous simulations have sug-gested that general GBs can absorb dislocations by meansof local changes in the GB structure [26]. It can be expectedthat such a mechanism would become more and more dif-ficult when the number of dislocations involved in the plas-tic deformation process increases [42], therefore increasingthe number of stress intensities via a long-range stress fieldthat contributes to the rms strain and to the WH anisot-ropy. Such an effect explains why a fully reversible peakbroadening is observed after unloading of a tensiledeformed ED Ni, whereas after 25% strain a non-reversiblebroadening of the 311 peak is observed, and still no dislo-cations in the grain interiors can be observed by TEM.

The present work does not consider the effect the pres-ence of impurities within the nc structure has on the rmsstrain. Since experimental nc samples contain significantamounts of impurities such as C, H, Co and S [46,47],which may either remain within the grain interior or segre-gate to the GB region, such an effect needs to be investi-gated and will constitute future work in the simulation ofXRD spectra of computer-generated nc samples.

In conclusion, by introducing a lattice dislocation ortwin content into computer-generated nc samples and cal-culating the corresponding two-theta X-ray diffractionspectra, the peak integral-width anisotropy as a functionof scattering angle is found to reflect the experimentalanisotropies observed in as-prepared nc-Ni and nc-Cu.The simulations also demonstrate that a similar anisotropyin the WH plot as that induced by lattice dislocations canbe obtained by the presence of highly localized stress inten-sities in GBs induced by ‘‘absorbed’’ lattice dislocations.The results explain the WH anisotropy of as-preparedED Ni, its behaviour after deformation to large strainsand how this is has to be understood in terms of the lackin dislocations observed in grain interiors during TEMobservations.

Acknowledgements

The authors acknowledge the financial support of theSwiss-FN (Grant Nos. 2100-065152 and 200020-103714/1) and the European Commission (FP7-NANOMESO)for financial support.

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