Strain mapping along Al–Pb interfaces.pdf

7/27/2019 Strain mapping along AlPb interfaces.pdf

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Strain mapping along AlPb interfaces

Harald Rosner a,*, Christoph T. Koch b, Gerhard Wilde a

a Westfalische Wilhelms-Universitat Munster, Institut fur Materialphysik, Wilhelm-Klemm-Strasse 10, D-48149 Munster, Germanyb Max-Planck-Institut fur Metallforschung, Heisenbergstr. 3, D-70569 Stuttgart, Germany

Received 11 May 2009; received in revised form 28 August 2009; accepted 30 August 2009Available online 28 September 2009

Abstract

Local strains at particlematrix hetero-interfaces in an AlPb composite produced by high-energy ball-milling were measured bymeans of geometric phase analysis (GPA) on the basis of high-resolution transmission electron microscope (HRTEM) images. Athrough-focus series of 10 experimental micrographs was used to retrieve the complex exit-face wave function and subsequently to cal-culate the components of the strain tensor using Pb as the reference lattice. The strain analysis revealed considerable strain for the inter-face regions with compressive strain for the Pb side and tensile strain for the Al side. The strained zones extend up to 23 lattice spacingson either side of the interface. The interface regions are not sheared or significantly rotated, which shows that the whole state of theinterface is elastically strained but not plastically deformed. A new possible interpretation of the melting behaviour of the Pb particlesin terms of the strains is discussed. 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Misfit dislocation; Strain mapping; Size-dependent melting; Focus series reconstruction

1. Introduction

Internal interfaces such as phase or grain boundaries arethe joints of materials and can thus be regarded as weakelements of the microstructure limiting e.g. the mechanicalproperties of a given material. One of the key parameters inthis regard is the stress located at these sites [1]. Our currentknowledge of interface stresses originating from internalinterfaces or boundaries in solids and their influence onthe materials properties is incomplete. Therefore, a betterunderstanding of stresses and strains in metallic systems

and semiconductor hetero-structures has become a majorresearch topic in materials science with special emphasison reduced dimensions in the case of applications ofnano-materials. One such fundamental aspect, where inter-faces play a dominant role, is the melting of confined sys-tems. Binary monotectic AlPb, which has been used formore than twenty years now [2], is an adequate referencesystem for this purpose [35]. Former transmission electron

microscopic (TEM) investigations revealed that the metal/metal interfaces display an epitaxial orientation relation-ship (cube-on-cube) even though there is a large latticemismatch of 22% between Al and Pb [25]. The accommo-dation of the lattice misfit was initially puzzling due toMoire fringes that veiled a direct observation of the inter-face structure [6]. Dahmen and Johnson suggested a model[7] that explained the accommodation of this large misfit inthe absence of a visible localization into interfacial dislo-cations by a combination of lattice vacancies and Pb parti-cles with so-called magic sizes for which the residual

misfit energy is minimized, i.e. for inclusion/matrix sizeratios of about 4/5. Recently, misfit dislocations werefound by high-resolution transmission electron microscopy(HRTEM) of uncovered Pb inclusions [8,9]. This was alsoconfirmed by high-resolution high-angle annular dark-fieldscanning transmission electron microscopy (HAADF-STEM) imaging of Pb inclusions located in the thicker partof the TEM sample [10].

This present investigation uses geometric phase analysis(GPA) [11] to investigate the strain distribution alongAlPb interfaces. Strain maps displaying the symmetric

1359-6454/$36.00 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.actamat.2009.08.065

* Corresponding author.E-mail address: [email protected] (H. Rosner).

www.elsevier.com/locate/actamat

Available online at www.sciencedirect.com

Acta Materialia 58 (2010) 162172
http://dx.doi.org/10.1016/j.actamat.2009.08.065mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.actamat.2009.08.065


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strain tensor components [11,12] were generated using Pbas the reference lattice. The interface regions reveal consid-erable strain and exhibit strain gradients extending up to 23 lattice spacings on either side. This analysis gives newinsight in the understanding of AlPb interfaces at whichno elastic distortions have been observed so far.

2. Experimental

The AlPb samples used in this study were produced byhigh-energy ball-milling as described in Ref. [13]. Thin elec-tron-transparent foils of as-milled material were preparedby low-angle (4) ion-milling (beam energy 3.0 keV) usinga PIPS (Gatan). TEM investigations were performed inan FEI Tecnai F20 G2 (field-emission gun, super-twin lens,Cs = 1.2 mm) operated at 200 kV. Since the spherical aber-ration of the objective lens leads to delocalization of theinformation [14,15] through-focus series of typically 20images were acquired. The individual high-resolution

micrographs were aligned and stacked. A flux-preservingiterative computer algorithm for the reconstruction of focalseries was used [16], in which the defocus Df, the sphericalaberration coefficient Cs, and the twofold astigmatism werecorrected. In this study 10 images with defocus values rang-ing from 154 to 203 nm were taken as input to retrievethe complex-valued exit-face wave function.

The AlPb was oriented along the [0 1 1] zone axis. Forthe acquisition of the micrographs a 1 k 1 k charged-cou-pled device camera (Gatan) was used. The method usedhere for measuring and mapping strain fields fromHRTEM images is the GPA developed by Hytch et al.

[11]. Within the GPA method, any displacement in theFourier-filtered local image pattern relative to a referencelattice image is interpreted in terms of a shift of atomicpositions. The strain tensor can then be obtained fromthe local structural displacements by numerical differentia-tion. A modified version which is based on the formalism inRef. [11] but allowing the geometric phase to be extractedfrom the complex-valued exit-face wave function wasimplemented in Digital Micrograph (Version 1.71.38,Gatan) [17] to calculate the components of the symmetricstrain tensor, e.g. exx, exy, eyy and the rigid-body rotationxxy [11,12], with Pb as the reference lattice. Circular maskswith radii producing a lateral resolution of 1 nm in the geo-metric phase images were set around the [1, 1, 1] and[1, 0, 0] reflections of the Pb lattice. Strain maps were cal-culated with respect to this reference lattice defined byg1 = [1, 1, 1]

Pb and g2 = [1, 0, 0]Pb. A cosine function

which produced a gradual transition of the mask valuefrom 1 to 0 within the outer 20% of the mask radius wasused for smoothing the edges of the circular masks. Itshould be noted that image distortions introduced by theprojector lens [18] or the imaging filter (Gatan GIF 200)as well as delocalization due to coherent aberrations ofthe objective lens are negligible because the complex-valuedexit wave function retrieved from focus series reconstruc-

tion was used [19].

3. Analyses and results

In this study our attention is focused on local strainsmanifested at a PbAl particlematrix heterophase-inter-face. The TEM investigations were performed in [0 1 1] ori-entation. Thus the interfaces are parallel to the electron

beam. The Fourier transform given as inset in Fig. 1a con-firms an epitaxial orientation relationship (cube-on-cube)between the Al matrix and the Pb nanoparticle.

To exclude misinterpretations related to Moire patternsor buried structures, uncovered Pb inclusions exhibitinghetero-structures with the Al matrix, as shown in Fig. 1a,were sought [8,9]. Fig. 1a shows an example of an imagefrom the through-focus series (image No. 2). The defocusvalue of this micrograph was determined to beDf= 156 nm by analysing the zero-crossings in the con-trast transfer function of this image. All other defocus val-ues were refined by the reconstruction algorithm [16]resulting in a mean defocus step of5.86 nm between suc-

cessive images. Fig. 1a contains two distinct interfaces, a[1, 1, 1] as well as a [1 0 0], on two lateral sides of thePb nanoparticle with the Al matrix. The aspect ratio ofthe two facets (1 1 1)/(1 0 0) is 1.08. Misfit dislocationsoccurred statistically on every 5.5 Al plane (55 Al{1 1 1}-planes match 45 Pb {1 1 1}-planes including 10misfit dislocations), which accommodates exactly the nom-inal misfit of 22.2%. This fact is of importance since itshows that, although most of the Al matrix was removedaround the Pb particle, the original state, as for fullyembedded nanoparticles [10], is maintained.

Two representative Burgers circuits, drawn around a

misfit dislocation on each interface, exhibit closing failuresof the type a0

201 1 for the (1 0 0) interface and a0

42 11 for

the (1, 1, 1) interface. The latter closing failure would cor-respond to a compact 60 a0

2h1 1 0i dislocation forming

hexagonal network [8,20,21].The result of the focus series reconstruction was the

retrieval of the complex-valued exit-face wave functionwhich was further used for calculating the strain maps.The retrieved phase and amplitude are shown in Fig. 1band c, respectively. The phase image (Fig. 1b) shows local-ized dislocation cores occurring periodically at the particlematrix interface. A simplified picture using Fourier filteringto enhance the dislocation cores is shown in Fig. 1d. Theslight waviness of the interface, which appears to be morepronounced for the {1 0 0}-interface, is readily apparentby the lateral offset of the individual dislocation cores fromeach other.

To point out the strained state of the particlematrixinterface, bright features in the phase image have beenassociated with Al (blue) and Pb (red) atomic columns(Fig. 2a). Matching peak positions of HRTEM images orphase images with the positions of atomic columns is thebasis of all currently used strain mapping algorithms oper-ating in real space, such as the peak finding [2225] andpeak pairs [26] algorithms (see Ref. [27] for a comparison

of peak finding and GPA).

H. Rosner et al. / Acta Materialia 58 (2010) 162172 163


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An unrelaxed model interface (Fig. 2b) was constructedusing two rigidly coupled lattices reflecting the mismatchof Al and Pb. At the intersection of the {1 0 0}/{1 1 1}-inter-face, a few Al columns had to be removed deliberately tomake a more direct association with the experimental situa-tion(Fig. 2a) possible. This led to the formation of an atomicstep of 9 Al columns in length. The interpretation of the

phase image in terms of atomic column species and position

is obviously not unique. However, the presented modelappears to be the most likely interpretation of the data.

This unrelaxed model (Fig. 2b) has been used to gener-ate the reference strain maps shown in Figs. 3b6b, whichshould reflect the state of faceted Pb inclusions embeddedin Al neglecting any interaction between the two lattices.

Strain maps were calculated, as described in the experi-

mental section, for the experimental image as well as for

Fig. 1. (a) High-resolution micrograph (No. 2 of the through-focus series) at a defocus value of156 nm showing a particlematrix hetero-interface of AlPb projected along the [0 1 1] zone axis. Two interfaces, a [1, 1, 1] as well as a [1 0 0] (top), remain on two lateral sides of the Pb nanoparticle, which islocated at the very thin edge of the foil (note the amorphous nature of the other sides). The inset displays the Fourier transform revealing an epitaxialorientation relationship (cube-on-cube) between the two lattices. Representative Burgers circuits, drawn around a misfit dislocation on each interface,exhibit closing failures of the type a0

201 1 (top interface) and a0

42 11 (left interface). 10 such images of the through-focus series were used as input to

retrieve the complex-valued exit-face wave function. (b) Phase image retrieved from the through-focus series using 10 images with defocus values rangingfrom 154 to 203 nm as input. (c) Amplitude image retrieved from the through-focus series using 10 images with defocus values ranging from 154 to203 nm as input. (d) Filtered phase image showing the (1, 1, 1) lattice fringes only. The misfit dislocations are clearly visible. Note the lateral offset ofthe individual dislocation cores from each other.

164 H. Rosner et al. / Acta Materialia 58 (2010) 162172


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the model interface shown in Fig. 2b. The correspondingresults, a juxtaposition of experimental (a) and model inter-face (b) in each case, are depicted in Figs. 36. Conse-quently, each strain map of an experimental image isdirectly comparable with the corresponding strain map ofthe model interface, and significant differences between

strained and unstrained states can be determinedimmediately.

The strain analysis reveals considerable strain (greencolour) around the interface regions extending up to 23lattice spacings on either side. Compressive strain is foundfor the Pb side and tensile strain for the Al side. Strain

Fig. 3. (a) Experimental mean dilatation (trace of the strain tensor: exx + eyy) using Pb as the reference lattice. Strains ranging from 40% to + 40%are displayed. The Al matrix is displayed in blue colour, indicating a strain state of 22%. The Pb particle is depicted in a mixture of green/red,representing the zero strain level (reference lattice). A diffuse greenish zone along the interface region is discerned, indicating considerable strains. Note theoffset position of the individual dislocation cores at the interfaces and the curvature (bulging nature) of the mean dilatation between them marked byarrows. (b) Mean dilatation (trace of the strain tensor: exx + eyy) using the model interfaces shown in Fig. 2b. Strains from 40% to + 40% aredisplayed. Note that the step in the model ( Fig. 2b) evokes considerable extra strain around the bend (indicated by the arrow), which is not observed in the

experimental strain map shown in (a).

Fig. 2. (a) Phase image superimposed with a possible atomic model consisting of Al (blue) as well as Pb (red) atom columns. The positions of the atomiccolumns have been adjusted to roughly match those of the local maxima in the phase image. (b) Model interface (unrelaxed) constructed by two rigidlycoupled lattices reflecting the mismatch of Al and Pb. Moreover, an unrelaxed atomic step was deliberately integrated around the intersection of the{1 1 1}/{1 0 0} interface. This model interface was used as a reference to compare to the experiment situation. Al columns are blue and Pb columns are red.



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peaks (hot spots) originating from the interfacial disloca-tion cores are observed, accommodating the misfit of22% between the two lattices. The general appearance of

the interfaces in terms of strains is rough and wavy, whichmanifests a non-equilibrium state of the particlematrixinterface of ball-milled material. In the following a detailed



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strain analysis based on the strain maps of the individualstrain tensor components is presented.

Fig. 3 shows the mean dilatation (trace of the strain ten-sor: exx + eyy) which summarizes the two individualstrain tensor components exx and eyy in one image. The fac-tor 0.5 was used to display the matrix and particle on the

same strain level; that is, 22% (blue colour) for the Almatrix and zero (reference lattice) for the Pb particle (mix-ture of red and green).

First of all, the wavy nature of the experimental inter-face compared to the model (c.f. the green line in Fig. 3aand b) is striking. The step, which was deliberately incorpo-rated in the model to create the best match to the experi-mental interface, appears clearly in the strain map of themodel (c.f. arrow in Fig. 3b) as a bulge where the curvatureis more extreme than between the other dislocation cores.Interestingly, this step is not observed in the experimentalstrain map (Fig. 3a). This means that this particular atomicstep must be relaxed at the particlematrix hetero-interface.

However, several other bulges are seen between the disloca-tion cores for the {1 1 1} as well for part of the {1 0 0}-interface in the experimental strain map, indicating thelocal roughness of the interfaces. These are marked bywhite arrows on Fig. 3a. It is further noticed that the dislo-cation cores (hot spots) show a lateral offset from eachother, supporting the rough and wavy character of theinterface regions. A detailed strain analysis for the{1 1 1}-interface, which is representative of both types ofinterfaces, is presented in the following: the misfit disloca-tion cores show a compressive part for the Al side and ten-sile strains for the Pb side (Fig. 4a). This situation reflects

the extra Al plane (5 Al planes meet 4 Pb planes). Theintermediate region between the dislocation cores exhibitsconsiderable strains along both sides of the interfaceappearing in a diffuse green colour (Fig. 4a). Since GPAallows us to calculate the strain values for each pixel, astrain profile was measured across the {1 1 1}-interfacealong the length and averaged over the width of the boxin Fig. 4a. A strain gradient is discerned extending up to3 lattice spacings on both sides (Fig. 4c). Bearing in mindthat the zero strain level is the equilibrium level for thePb particle and 22% (blue colour) displays the equilib-rium state of the Al matrix, compressive strain is identifiedfor the Pb side of the interface region whereas the Al side isunder tension. This finding is now compared with themodel {1 1 1}-interface shown in Fig. 4b. The generalappearance of the model interface region is much

smoother. Again, a strain profile was measured across theinterface along the length and averaged over the width ofthe box in Fig. 4b. Here (Fig. 4d) the strain gradientextends over 0.7 nm in total only roughly half of theextent of the experimental gradient and is consistent withthe mask sizes (radius 1 nm1 in reciprocal space) used to

generate the strain maps. For this reason, the pronouncedstrain states of the experimental interface regions are ascer-tained. Similar results were found for the {1 0 0}-interfaceregion which are not graphically represented.

Fig. 5 compares the shear components of the strain ten-sor. Strain peaks are located along the particlematrixinterface matching with the position of the misfit disloca-tion cores. It is apparent that the dislocation cores arealigned in different ways, i.e. parallel at the {1 1 1}-inter-face and perpendicular at the {1 0 0}-interface. This reflectsthe different character of the misfit dislocations, namely 60dislocations at the {1 1 1}-interface and pure edge disloca-tions at the {1 0 0}-interface. Compression strains are

found on the Al side reflecting the presence of the extraplane compared with the Pb side, which is under tension.The analysis shows that the shear strain originates fromthe dislocation cores, and that the interface regions inbetween the dislocation cores (hot spots) do not showappreciable shear strains. However, pre-located shearstrain zones (indicated by black arrows on Fig. 5a) in theAl matrix near the {1 0 0}-interface are noticeable. Becauseof their distance from the interface, they are categorized asdefects of the Al matrix having minor importance for theinterface region itself. Moreover, it is noted that the delib-erately incorporated step in the model interface at the inter-

section of the {1 1 1} and {1 0 0} interfaces (Fig. 2b) led tonoticeable shear strains (Fig. 5b), which are not observed inthe experimental case shown in Fig. 5a.

The strain analysis is completed by looking at the rigid-body rotation maps presented in Fig. 6. Again, the disloca-tion cores are clearly observable as hot spots. They are allaligned parallel with the interfaces, some appearing slightlytwisted and asymmetric compared to the model interface(Fig. 6b). Certain areas (indicated by white arrows) alongthe experimental interface in Fig. 6a appear slightly rotated(60.1). The accuracy of the rotation measurement isabout 0.1 [12]. However, the artificial step in the model(Fig. 6b) is not found in the experimental interface(Fig. 6a). Again, pre-located zones (indicated by blackarrows) in the Al matrix near the [1 0 0] interface arenoticeable but extraneous to the interface region.

Fig. 4. (a) Enlarged view of the strain map (mean dilatation) of the experimental {1 1 1}-interface region shown in Fig. 3a, revealing details of: (i) thestrained state (diffuse green colour) and, (ii) the dislocation cores (four hot spots). The colour legend is the same as shown in Fig. 3a. (b) Enlarged view ofthe strain map of the model {1 1 1}-interface region shown in Fig. 3b. (c) The strain profile was measured across the interface along the length andaveraged over the width of the box shown in (a). A strain gradient is seen across the interface region starting from the Al matrix (equilibrium strain level is22%) into the Pb particle (reference strain level is zero) revealing compressive strain for the Pb side whereas the Al matrix shows tensile strain. The totallength of the gradient is 1.5 nm. (d) The strain profile was measured across the model interface along the length and averaged over the width of the boxshown in (b). A strain gradient is obtained across the interface region starting from the Al matrix (equilibrium strain level of22%) into the Pb particle(reference strain level of 0%). Again, compressive strain is found for the Pb side whereas the Al matrix shows tensile strain. Note the difference in length:

here the gradient extends about 0.7 nm whereas in the experimental case it extended about 1.5 nm.

3



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Fig. 5. (a) Experimental shear component exy using Pb as the reference lattice. Shear strains on a scale from 40% to +40% are displayed. Pre-locatedshear strain zones (indicated by black arrows) in the Al matrix near the [1 0 0]-interface are noticeable. (b) Shear component exy using the model interfacesshown in Fig. 2b. Strains on a scale from 40% to +40% are displayed. Note the increased shear strain at the intersection of the {1 1 1} and {1 0 0}interfaces originating from the deliberately incorporated step in the model interface (Fig. 2b), which is not observed in the experimental case shown in (a).



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Fig. 6. (a) Experimental rigid-body rotation xxy using Pb as the reference lattice. Rotations ranging from 0.4 to +0.4 (anticlockwise positive) aredisplayed. Certain areas (indicated by white arrows) along the interfaces appear slightly rotated (6 0.1). (b) Rigid-body rotation xxy using the modelinterfaces shown in Fig. 2b. Rotations from 0.4 to +0.4 are displayed.



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4. Discussion

The detailed strain analysis presented in the last sectionunambiguously shows considerable strain for the interfaceregions of PbAl particlematrix hetero-structures withcompressive strain for the Pb side and tensile strain for

the Al side. The strained zones extend up to 23 latticespacings on either side of the interface. The interfaceregions are not sheared or significantly rotated, which leadsto the conclusion that the whole state of the particlematrixinterface is elastically strained but not plastically deformed.Thus, based on the comprehensive strain analysis presentedhere, a new interpretation is derived for particlematrixhetero-interfaces of ball-milled AlPb composites. Below,the strain analysis is critically discussed with respect to itsrobustness and with respect to the melting behaviour ofnanometre-sized Pb inclusions embedded in the Al matrix.

4.1. Robustness

The robustness of GPA itself has been demonstratedpreviously in several case studies by comparison with otherstrain mapping algorithms [2227] and theoretical consid-erations [28]. The fact that the complex-valued exit-facewave function retrieved from focus series reconstructionis used here excludes further errors due to image distortionsintroduced by the projector lens system [19]. Especially,delocalization due to coherent aberrations of the objectivelens is minimized [14,15]. However, in order to evaluatecorrectly the newly derived situation at the interfaceregions in terms of strains, each experimental strain map

was compared with a model interface constructed by tworigidly coupled lattices reflecting the mismatch of Al andPb accurately. Moreover, an unrelaxed atomic step wasdeliberately integrated around the intersection of the{1 1 1}/{1 0 0} interface. This artificially created step wasclearly discerned as a bulge in the strain map of the modelinterface (Fig. 3b). However, bulges were found in theexperimental maps at positions where the model did notshow any (Fig. 3a). The observation of significant bulging,i.e. the deviation of the strain field from planar geometryresembling an edge-on interface indicates that the modelbased on two unrelaxed lattices meeting at a planar inter-face is too simplified to resemble the experimental configu-ration. Thus, the current result indicates that the topologyof the real interface between Pb and Al entails corrugationsalong the direction of the incident electron beam, i.e.mixed atomic columns. The fact that the features pro-duced by the step in the artificial interface are not foundin the strain maps of the real particle indicates that in-planedefects, such as steps, do not occur on the interface of theexperimentally analysed particle. However, the step in themodel can be used as a quantitative measure to achieve aproper evaluation of the particlematrix hetero-interfacein terms of strain. In addition, the strain profile acrossthe model interface reveals an interface width of 0.7 nm

in total, which is mainly determined by the radius of the

mask used in the reciprocal space to generate the geometricphase maps. However, this value would also fit with thestructural and diffusional width of grain boundaries/inter-faces of about 2 * a0 % 0.50.8 nm (a0: lattice parameter)[2932].

4.2. Rough and wavy interfaces

The experimental situation of the interface region isdepicted in Fig. 3a and in more detail in Fig. 4a, wherebulges and dimples are found. It is noted that the{1 1 1}-interface appears to be flatter (general curvatureof the whole interface) than the {1 0 0}-particlematrixinterface. The significance of this observation is uncleardue to the fact that only one interface of each kind wasanalysed so that any statistical basis is missing. However,based on the specific excess free energy densities of thetwo interfaces, a flatter {1 0 0}-particlematrix interfacewould be expected since the excess interface free energy

density of that interface is higher.Intrinsically, such diffuse interfaces need to be involved

in any transient state that develops via atomic motiontowards a more stable configuration based on faceted inter-faces. The lifetime of such transient states depends on tem-perature and might be extended in time. The observation ofpartial faceting after annealing severely deformed AlPbcomposites indicates that such microstructures are certainlytransient in nature [13,33]. However, whether the atomic Aland Pb columns are still separated or if atomic mixing isinvolved here cannot be determined on the basis of the strainmapping as presented in this study.

4.3. Observation of compressive strain at the interface

regions of the Pb particle

The detailed strain analysis presented in the precedingparagraphs has clearly shown that the interface regionsof PbAl particlematrix hetero-structures are elasticallydistorted with compressive strains for the Pb and tensilestrains for the Al side. Thus a pressure or strain relief atthe interface accommodated by vacancies or creep cavita-tion has to be excluded [3436]. However, an earlier onsetin melting of {1 0 0} interfaces [37] cannot be explained interms of the strains observed, since the strain state wasfound to be similar for both types of interfaces, i.e.{1 0 0} and {1 1 1}. Moreover, the interface regions arenot sheared or significantly rotated, which leads to the con-clusion that the whole state of the particlematrix interfaceis elastically strained but not plastically deformed. Thus,this finding actually shows the validity of early two-phasemodels that describe nanostructured materials in terms ofa core/shell approach [38]. In addition, this observationstresses the importance of the interface contributions withrespect to property tuning of nanostructured materialssince bulk material could not support such large deviationsfrom equilibrium without rapid relaxation via the genera-

tion of localized and highly mobile defects.



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4.4. Link to the melting behaviour of matrix-encased

nanoparticles

To link the microstructural observation together withthe strain analysis to the melting behaviour of this individ-ual Pb particle is both desirable and tempting but it has to

remain speculative since the direct melting was not moni-tored. However, a couple of statements can be made andare discussed below.

First of all, to judge the situation of the analysed Pb par-ticle embedded in an Al matrix correctly, it should beemphasized that the AlPb composite used for this studywas produced by high-energy ball-milling [13]. The meltingbehaviour of Pb particles in such a material is usually char-acterized by a melting point depression to temperaturesbelow the melting point of bulk Pb [3,13,39]. The morphol-ogy of the embedded Pb inclusions was recently determinedby electron tomography [40] and two different species werefound. The majority of particles had curved spherical

shapes with some degree of faceting in their outline. About4% of Pb particles were cuboctahedrally shaped, displayingclear {1 0 0} and {1 1 1}-facets and the Pb particle investi-gated in this study belongs to this latter (minor) species.However, it has been previously shown for such compositesthat annealing led to faceting of the Pb particles which sub-sequently showed an increased melting temperature, Tm,above the melting point of bulk Pb [13]. Thus, it is reason-able to expect that the Pb particle analysed here wouldshow a certain increase of the melting point (up to 13 Kafter [13]). Such increase of Tm can be described by theCouchmanJesser equation [41]:

Tm T0 1 3clm csm

r L

1

Tm is the melting point of the embedded particle and T0is the bulk melting point. clm and csm are the liquid/matrixand solid/particle interfacial energies, L is the latent heatand r the particle radius. Pressure effects or strains due todifferential thermal expansion of the matrix and the embed-ded phase are not included in this expression. However, thestrain analysis of the Pb particle presented here unambigu-ously reveals considerable strain at the interface which hasto be taken into account. An adequate way to describe themelting behaviour (first-order transition) consideringstrain/pressure is given by the Clapeyron equation:

DV

DS Dp DTM 2

DV, DS, Dp, DTare the changes in volume, entropy, pres-sure and melting temperature, respectively. Therefore anyincrease in pressure would shift the melting point to highertemperatures. Since the outer regions of the Pb particle areunder compressive strain (negative) of large magnitude, ahigher melting temperature is expected according to Eq. (2).

The strain peaks (hot spots) in the strain maps originat-ing from the misfit dislocation cores at the interfaces do not

have an influence on the melting behaviour (depression or

superheating) [39] since the misfit is accommodated in thesame manner for both species of Pb particles [6,8], i.e.spherical or faceted.

However, it has been reported that steps/ledges at parti-clematrix interfaces lead to melting point depression [42].Thus, two facts could account for an increase of the melting

point, namely the faceted shape or the compressive strainaccording to the Clapeyron Eq. (2), whereas the occurrenceof steps could account for a decrease. Currently, it cannotbe said how these three contributions add up. More exper-imental facts in terms of strain mapping of curved particlematrix interfaces as well as a direct observation of the melt-ing of such excavated Pb particles have to be awaited beforeany further conclusions can be arrived at.

5. Summary

The comprehensive strain analysis presented in thisstudy provides new insight into particlematrix hetero-

interfaces of AlPb composites at which no elastic distor-tions have been observed before. We found considerablestrain for the interface regions with compressive strainfor the Pb side and tensile strain for the Al side. Thestrained zones extend up to 23 lattice spacings on eitherside of the interface. The interface regions are not shearedor significantly rotated, which leads to the conclusion thatthe whole state of the particlematrix interface is elasticallystrained but not plastically deformed. Since the outerregions of the Pb particle are under compressive strain oflarge magnitude, a higher melting temperature is expectedaccording to the Clapeyron equation, whereas the occur-

rence of steps at the interface could account for a decrease.However, further experimental facts are needed before theconsequences of the strain analysis can directly be linked tothe melting behaviour of matrix-encased Pb particles.

Acknowledgment

CTK acknowledges financial support by the EuropeanCommission under contract No. NMP3-CT-2005-013862(INCEMS).

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Strain mapping along Al–Pb interfaces.pdf

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