Shapes and sizes of nanoscale Pb inclusions in Al

Materials Science and Engineering A304–306 (2001) 187–193

Shapes and sizes of nanoscale Pb inclusions in Al

E. Johnsona,∗, A. Johansena, U. Dahmenb, S. Chenb, T. Fujii b

a Ørsted Laboratory, University of Copenhagen, Universitetsparken 5, DK-2100 CopenhagenØ, Denmarkb National Center for Electron Microscopy, LBNL, UC Berkeley, Berkeley, CA, USA

Abstract

Al–Pb alloys are monotectic and characterized by a large miscibility gap in the liquid phase area and extremely limited mutual solubilityin the solid phase. Due to the extent of the miscibility gap the alloys are difficult to make in conventional processing. However, alloyswith relatively homogeneous microstructures of fine Pb inclusions in an Al matrix can be made by metastable processing such as rapidsolidification, ion implantation, ball milling and physical vapor deposition.

The first two techniques have been employed to make alloys of Al with 0.5–3 at.% Pb. The alloys contain fine dispersions of nanoscale Pbinclusions with sizes in the range from 1 to about 20 nm after ion implantation and from about 10–500 nm after rapid solidification. Inclusionsembedded in the Al matrix are single crystalline, and they grow in parallel cube alignment with the matrix. They have cuboctahedral shapewith atomically smooth1 1 1 and1 0 0 facets determined from a minimization of the interface energy. Using high resolution TEM,two types of deviations from the classical Wulff construction which alter the shape of the inclusions, have been studied. The smallestinclusions, less than about 20 nm in size, adopt a series of magic sizes that can be related to the occurrence of periodic minima in theresidual strain energy. Likewise, in this size range, the energy contribution from the cuboctahedral edges becomes non-negligible leadingto an increase in the aspect ratio of the inclusions with decreasing size.

Inclusions located in grain boundaries in general adopt a single crystal morphology where one part is faceted and grows in parallel cubealignment with the matrix grain, while the other part has a shape approximating a spherical cap. In special cases such as twin boundariesand1 1 1 twist boundaries, the inclusions are bicrystalline where each part is aligned with the respective grain and the two parts areseparated by a boundary similar to that of the matrix. These shapes can be explained using the Cahn–Hoffmanξ -vector construction, whichgeneralizes the Wulff construction to determine equilibrium shapes at anisotropic interfaces and their junctions. © 2001 Elsevier ScienceB.V. All rights reserved.

Keywords:Electron microscopy; Al alloys; Equilibrium shapes

1. Introduction

For the past two decades structural, thermodynamic anddiffusion properties of crystalline particles completely em-bedded in a metallic matrix have been investigated in detailin a variety of heterogeneous alloys of immiscible elementsproduced by both ion implantation and rapid solidification.The monotectic binary Al–Pb alloy system has a largemiscibility gap in the liquid phase and extremely limitedmutual solubility in the solid phase, and it has been used asa key-type system for such investigations. In an early workby Thackery and Nelson [1] it was demonstrated that ionimplantation of Pb into Al led to formation of fine distribu-tions of nanoscale Pb precipitates or inclusions which grewin parallel cube alignment with the Al matrix. Roth et al.[2] studied conventionally cast Al–Pb alloys with specialemphasis on grain boundary inclusions and Grugel and

∗ Corresponding author. Fax:+45-3532-0460.E-mail address:[email protected] (E. Johnson).

Hellawell [3] used Al–Pb alloys to study casting throughthe miscibility gap. In a series of more recent experimentsrapid solidification has been used to make Al–Pb alloys withdense distributions of nanosized Pb inclusions with sizesaround 20–500 nm [4–9], while ion implantation was foundto produce inclusions with about an order of magnitudesmaller sizes [10–15]. In all these experiments nanosizedf.c.c. Pb inclusions were found to grow in parallel cubealignment with the Al matrix despite a mismatch of 22%between the two f.c.c. structures. The Pb inclusions havethe same structure as the Al matrix and their shape wasalways found to be cuboctahedral with large smooth1 1 1facets and smaller1 0 0 facets.

As the inclusion size decreases a larger fraction of Pbatoms will be located in the Al–Pb interfaces with a localenvironment that is different from atoms located in the bulk.This will inevitably influence both the physical and struc-tural properties of nanoscale inclusions which have beenfound not only to deviate strongly from bulk behavior butalso to display a strong size dependence. The low-energy

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188 E. Johnson et al. / Materials Science and Engineering A304–306 (2001) 187–193

nature of the interfaces has a profound influence on thethermal properties of the inclusions. Melting is associatedwith a distinct superheating [4–6,13,14] that increases in-versely with inclusion size [14] and can be as large as 100 Kabove the bulk melting point for the smallest inclusionswith sizes of a few nanometers. Most recently, ball millinghas been used to produce alloys with Pb inclusions that arerandomly oriented in the matrix [16,17]. These inclusionshave random non-faceted high-energy interfaces, and ac-cordingly it has been found that they melt below the bulkmelting point with pre-melting temperatures of the order of15–20 K [16].

In the present paper we will review some of the morerecently discovered properties of nanoscale Pb inclusions inAl. In the absence of volume stress, the morphology of theinclusions is determined solely from an optimization of theshape with respect to interface energy. In this context, Her-ring [18] showed that the Wulff construction predicts a shapewhere the distance between opposite facet pairs scales withthe surface energy, and analytical expressions for cubic crys-tal shapes were obtained by Nelson et al. [19]. Replacing sur-face energies with interface energies and applying the Wulffconstruction to embedded Pb inclusions in Al, measurementsof facet pair distances thus provide a relative measure of theassociated interface energies [4,5,15]. Using high resolutionelectron microscopy, deviations from the ideal Wulff shapeswere found to occur for inclusion sizes below 10–20 nm. Aresidual strain energy term oscillating with inclusions size,was found to lead to a magic size effect with preferentialoccurrence of inclusion sizes for which the residual strainenergy term is minimal [20,21]. Extra energy terms fromthe edges between the smooth facets were found to cause areduction in the degree of truncation for the smallest inclu-sion sizes [22]. Inclusions located in Al grain boundarieswere found in general to be single crystalline with a shapethat can be explained in terms of a generalized Wulff con-struction based on the Cahn–Hoffmanξ -vector construction[23,24].

Fig. 1. Pb inclusions in Al made by ion implantation at 425 K: (a) conventional bright-field TEM micrograph from a (1 1 0) crystal showing inclusionsabout 5–10 nm in size; (b) associated SAD pattern demonstrating the parallel cube alignment of the inclusions; (c) the cuboctahedral shape of theinclusions with larger1 1 1 and smaller1 0 0 facets obtained from a 3D analysis.

2. Morphology of Pb inclusions in the Al matrix

Fig. 1a shows a conventional bright-field TEM micro-graph from an Al crystal implanted with Pb ions at 425 Kto a concentration of about 1 at.%. A diffraction analysisof the inclusions shows that they have f.c.c. structure andgrow in parallel cube alignment with the Al matrix (Fig. 1b).The images of the inclusions are characterized by a strongmoiré contrast originating from the 22% misfit between thetwo parallel f.c.c. lattices witha0(Al ) = 0.4048 nm anda0(Pb) = 0.495 nm, respectively. A 3D analysis of the in-clusions shows that their shape is cuboctahedral with larger1 1 1 facets and smaller1 0 0 facets (Fig. 1c). In the(1 1 0) orientation two pairs of opposite1 1 1 facets andone pair of opposite1 0 0 facets are seen edge-on, andthis orientation is therefore ideal for observation of the in-clusions (Fig. 1a). In zone-axis high-resolution micrographsthe lattice images are heavily obscured by the moiré con-trast (Fig. 2a) which makes it difficult to see both the in-clusion lattice itself and the shape of the inclusions. Imageswith better visual appearance can, however, be obtained foroff-axis orientations [25] where in particular the outline ofthe inclusions is more apparent (Fig. 2b).

The inclusions are not coherent because the lattice mis-match is too large to be accommodated elastically. Theyare not typically incoherent either because a well-definedorientation relationship is strictly maintained. The largemisfit of 22% between the two lattices will in general in-duce excessive volume strains in the matrix surrounding theinclusions. Nevertheless, evidence for such volume strainshas never been observed in bright-field or dark-field TEMimages which are very sensitive to strain variations. Duringin situ melting and solidification experiments where theinclusions undergo rapid shape changes, strain lobes char-acteristic of volume strains have, however, been observedto form and disappear again within a fraction of a second.The energy tentatively stored in the large volume strains arebelieved to be almost entirely relieved when the strains are

E. Johnson et al. / Materials Science and Engineering A304–306 (2001) 187–193 189

Fig. 2. High-resolution micrographs of Pb inclusions in Al demonstrating the cuboctahedral shape in the (1 1 0) orientation: (a) zone-axis micrographillustrating how the strong moire contrast complicates the detailed perception of the inclusions; (b) off-axis micrograph taken under conditions where theAl matrix lattice is nearly invisible provides an enhanced perception of the shape of the inclusion.

accommodated by lattice vacancies, which are formed ther-mally even at room temperature and which are in copioussupply above 500 K where the inclusions are equilibrated.This strain accommodation, which for a faceted inclusioncan only be achieved to the dimension of a single latticeplane, will be optimal whenever the match between the twolattices is perfect. Therefore, distortions of the interfaces arereduced to a minimum and the interfaces are smooth. Thehigh-resolution micrographs show that the facets of the in-clusions approach smoothness on atomic scale, confirmingthe earlier X-ray diffraction analysis of truncation rods fromthe confining aluminum cavities [14]. Due to the smoothnessand the minimal distortions the energies of the interfaces aregenerally considered to be low.

3. Deviations from the classical Wulffconstruction — magic sizes

In the absence of volume strains the equilibrium shapeof a crystal or an embedded inclusion is given by the clas-sical Wulff construction which is the shape that minimizesthe total surface or interface energy, respectively. For cubicinclusions bounded by1 1 1 and1 0 0 faces alone theWulff shape is an octahedron with1 1 1 facets and cornertruncation to produce1 0 0 facets [19]. The aspect ratioρ = c/a, defined as the ratio between opposite1 0 0 and1 1 1 facet pair distancesc and a, respectively, is thenequal to the ratio between the respective interface energiesγ 1 0 0/γ 1 1 1. As inclusions become smaller and approachnanoscale size, deviations from the simple Wulff construc-tion begin to gain importance. Two extra contributions canbe added to the interface energy, namely a residual strain

energy which induces a magic size effect [20], and a termfrom the contribution of edge energy which makes theaspect ratioρ size dependent [22].

Inclusion sizes were measured as the distance betweenpairs of opposite1 1 1 facets, e.g. the distance as shownin Fig. 3a. Measurements on high-resolution micrographscould be made with an accuracy of plus or minus one planarspacing when the spacing of the1 1 1 aluminum latticeplanes was used for calibration. Fig. 3b shows the size dis-tribution for inclusions in a sample that was equilibrated for1 h at 575 K, where the inclusion size is measured in units ofthe 1 1 1 aluminum planar spacing of 0.234 nm. It is im-portant to note that the histogram in Fig. 3b is constructedfrom all the measured facet pair distances without averagingover the two sets of facet pairs in each inclusion. The magicsize effect is seen as a preferred occurrence at regular inter-vals of certain sizes while sizes in between are less common.This effect, which is also known for free clusters and for 2Disland growth, can be explained in terms of a residual strainenergy that remains in the interfaces when volume strainsare accommodated by thermal vacancies. Using the simpleapproximation of a spherical inclusion in an isotropic ma-trix [20] the residual strain energyEres can be expressed interms of the oscillating misfitΩ between the two lattices as

Eres = 4π

3ARΩ2(R) (1)

where

Ω = minp|qdPb − pdAl | (2)

HereR is the inclusion size andA = (18µK)/(4µ + 3K)

andµ andK are the shear modulus and the bulk modulus,


Fig. 3. (a) High resolution micrograph of Pb inclusion in Al with indication of the facet distance a between an opposite1 1 1 facet pair; (b) histogramof measured1 1 1 facet pair distances displaying magic size behavior. The gray curve shows the residual strain energy obtained from Eq. (3). Notethat minima in the energy curve coincide with the frequently observed magic sizes.

respectively. The misfit functionΩ is given in terms of thelattice spacingsdPb anddAl of lead and aluminum, respec-tively, andp andq are running series of integers countingthe number of lead and aluminum planes that meet alongthe interface.Ω is an oscillating function of the inclusionsizepdAl and geometrically it represents the moiré patternbetween the two corresponding lattice distances. For thedistance between opposite1 1 1 facets the oscillation hasa wavelength of 1.28 nm equal to the corresponding moiréspacing. The peak value of the oscillations increases linearlywith inclusion sizeR, and hence it is only important for thesmallest inclusions, while the interface energy terms that in-creases withR2 will dominate for larger inclusion sizes.

Using an alternative approach, the nature of the resid-ual strain can be considered analogous to that of a “cage”of stair-rod dislocations along the edges of a stacking faulttetrahedron. A faceted inclusion with octahedral shape willthen be analogous to a stacking fault octahedron where thestacking faults are replaced by interfaces and the stair roddislocations are replaced by dislocations with variable Burg-ers given by the moiré function (Eq. (2)). Using the ex-plicit solution for the interaction between straight dislocationsegments given by de Wit [26], the residual elastic energycan then be calculated from a summation of the dislocationself-energies and all the pairwise interaction energies giving

Eel = Ω2R3µ

π(1 − ν)

ln

(R

r0

)− 0.43

(3)

wherer0 is the core radius of the dislocation. Eq. (3) showsthat the residual elastic strain energy in this model is alsoproportional toΩ2 and hence oscillates with size with a pe-riodicity given by the moiré function. The energy of a leadinclusion in aluminum can then be described by the sum ofan interface energy and a residual misfit energy which in

turn is composed of two terms, one due to broken bonds andone due to elastic distortion. The bond energy term varieslinearly with size, while the strain energy term oscillates be-tween zero and a maximum that increases with size roughlyasR(ln R). The graph of the residual misfit energy calculatedfrom Eq. (3) is overlaid with the experimental size distribu-tion in Fig. 3a. There is a nearly perfect match between theexperimental results and the calculated curve, and it showsthat the magic sizes are those for which the residual misfitis smallest.

4. Deviations from the classical Wulffconstruction — changes in aspect ratio

Classically, the aspect ratioρ = c/a defines theanisotropy of the inclusions expressed as the ratioγ 1 0 0/γ 1 1 1 between respective interface energies. For larger Pbinclusions in Al the aspect ratio has been reported to beabout 1.15 [4,5], very close to the value of 2/

√3 = 1.16

calculated for a free crystal using a geometrical nearestneighbor, broken bond model. However, a significant in-crease to a value of 1.28 has also been observed for smallenough inclusions to display magic size behavior [15].Fig. 4a shows the1 0 0 and1 1 1 facet pair distancescanda used for measurements of the aspect ratio. In order tominimize the influence of magic size effects, the aspect ra-tio was always obtained as an average over the two1 1 1facet pair distances. Fig. 4b shows a plot of measured val-ues of the aspect ratioρ = c/a for Pb inclusions in Al asa function of inclusion size a for sizes in the range from 1to 55 nm. The data show a very large scatter that increaseswith decreasing inclusion size. This is not due to measure-ment errors which for the small size inclusions are of the


Fig. 4. (a) High resolution micrograph of Pb inclusion in Al with indication of the1 0 0 and 1 1 1 facet distancesc and a, respectively, used fordetermining the aspect ratio; (b) plot of the measured aspect ratios as a function of inclusion size. The full line shows the least-squares fit of the data toEq. (4). A series of inclusion shapes associated with various aspect ratios are shown on the second axis.

order of plus or minus one lattice spacing. It is a scatterthat originates from the tendency of the smaller inclusionsto retain magic size inter-facet distances when they do notcontain a perfect magic size number of Pb atoms. These in-clusions will adopt various asymmetric shapes giving largedeviations from the ideal aspect ratio (see, for exampleFig. 4a). A detailed statistical analysis of the data in Fig. 4b,based on binned data sets, shows that the increase in aspectratio for small inclusion sizes is statistically significant.This increase can be explained as a direct consequence ofthe contribution of edge energy to the total energy. For arelative edge energyτ = ε1 1 0/γ1 1 1, whereε1 1 0 is the en-ergy of a〈1 1 0〉 edge of unit length, the equilibrium aspectratio minimizing the combined effect of interface and edgeenergy can be calculated to be

c

a= ρ + √

6(τ/a)

1 + √2(τ/a)

(4)

The aspect ratio for larger inclusions is independent of sizeand approaches the value ofc/a = ρ = γ1 0 0/γ1 1 1. In con-trast, smaller inclusions have gradually less truncation witha limiting aspect ratio of

√3, the value for the untruncated

octahedron. This result has been anticipated earlier from nu-merical analyses [27,28], and agrees well with the presentset of data as seen from the two-parameter least-squares fitin Fig. 4b (full line) to the expression for the aspect ratiousingρ andτ as independent fitting parameters. The valuefor the “energy” parameters from the fit in Fig. 3 areρ =1.151± 0.005 andτ = (2.06± 0.16)a0(Pb), wherea0(Pb)

is the lattice parameter for Pb. The value forρ agrees wellwith the value found by Cantor and his colleagues [4,5], andit surprisingly close to the value of 2/

√3 = 1.16 obtained

from a geometrical nearest neighbor broken bond model.This model also predicts a value forτ = a0/

√6 = 0.41a0

which is about five times smaller than the measured value.

5. Morphology of Pb inclusions in Al grain boundaries

During both ion implantation and rapid solidification Pbinclusions form readily in Al grain boundaries. In generalthey are found to be larger than inclusions embedded in thematrix and a zone denuded of inclusions is often observed[29]. In most cases the inclusions adopt a single crystal mor-phology where one part is faceted and grows in parallel cubealignment with the matrix grain, while the other part has ashape approximating a spherical cap (Fig. 5a). BicrystallinePb inclusions with an internal grain boundary are only ob-served in special grain boundaries such as twin boundariesand pure twist boundaries (Fig. 6a). The overall shape of thegrain boundary inclusions may be understood qualitativelyin terms from a modified Wulff construction [23] based onthe Cahn–Hoffmanξ -vector construction [24] as the inter-section of two equilibrium Wulff shapes for the Pb inclusionacross the grain boundary (Figs. 5b and 6b). The Wulffshapes are the equilibrium shapes of the inclusions obtainedfrom the classical Wulff construction and drawn such thatrelative dimensions scale with the respective interface en-ergies. For aligned Pb inclusions in Al the Wulff shape is


Fig. 5. (a) High resolution micrograph of a Pb inclusion in a 90 〈1 1 0〉 tilt grain boundary in Al. The shape of the inclusion, which is single crystalline,can be understood in terms of theξ -vector construction (b) as an intersection between the two Wulff shapes. The distance between the two Wulff centersis equal to the grain boundary energy. The arrow indicates the position of a thin slab of Pb which is oriented parallel to the upper Al grain.

cuboctahedral while non-aligned inclusions are approxi-mated with a spherical Wulff shape where low-energy facetsare absent. As relative dimensions in theξ -vector plots scalewith various interface energies, the intersection shape defin-ing the morphology of a grain boundary inclusion is given bythe configuration where the distance between the centers ofthe two Wulff shapes scales with the grain boundary energy

Fig. 6. (a) Bright field micrograph of a series of Pb inclusions in an Al63 twin boundary. All the inclusions are internally twinned. The distancebetween the Wulff centers for the two cuboctahedral Wulff shapes in theξ -vector construction (b) is equal to the difference between the Al and Pb twinboundary energies. Note the appearance of reentrant edges on some of the inclusions.

(Fig. 5b). In the special case of twinned inclusions whichare bicrystalline, the distance between the two Wulff cen-ters is equal to the relative difference in the twin boundaryenergy between Pb and Al (Fig. 6b). Strictly, this construc-tion is valid only for inclusions where the two Wulff shapesmeet the grain boundary in a planar section [24]. When thisis not the case, such as for the Pb grain boundary inclusion


in Fig. 5, the intersection shape may be more complicatedor alternatively the otherwise straight grain boundary maytwist or pucker in the vicinity of the inclusions [30].

A frequently observed feature of the Pb grain boundaryinclusions is the appearance of a thin layer of varying thick-ness along certain parts of the rounded interface of the in-clusion (see arrow in Fig. 5a). This layer is a thin sheet ofPb oriented parallel to the adjoining Al grain. The orientedPb sheet is only a few atomic layers thick and it tends to bethe most pronounced for orientations near to the1 1 1Alplanes. It depicts a tendency for the general curved Al–Pbinterface to dissociate into an aligned Al–Pb interface andan internal Pb grain boundary. Such a dissociation, resem-bling the dissociation of certain twin boundaries in copper[31], will take place for orientations where the energy of therandom Al–Pb interface is larger than the combined energyof the faceted Al–Pb interface and an internal low energyboundary in the inclusion.

6. Conclusions

The cuboctahedral equilibrium shape of nanoscale Pbinclusions in Al can be understood from the classical Wulffconstruction. Despite a large misfit between the two latticesthe inclusions grow in parallel cube alignment with the ma-trix, they are free from internal defects, and they are sharplyfaceted on the1 1 1 and1 0 0 planes. However, as theinclusion size decreases, deviations from the classical Wulffconstruction become significant. The smallest inclusions,less than about 20 nm in size, adopt a series of magic sizesthat can be related to the occurrence of periodic minima inthe residual strain energy. Likewise, in this size range, theenergy contribution from the cuboctahedral edges becomesnon-negligible leading to an increase in the aspect ratio ofthe inclusions with decreasing size.

Inclusions located in grain boundaries have a compositemorphology generated from segments of two Wulff shapesjoined along the boundary. These shapes can be explainedusing the Cahn–Hoffmanξ -vector construction, which gen-eralizes the Wulff construction to determine equilibriumshapes at anisotropic interfaces and their junctions.

Acknowledgements

This work is supported by the Danish Natural Sci-ences Research Council and the Director, Office of Energy

Research, Office of Basic Energy Sciences, MaterialsScience Division of the US Department of Energy underContract No. DE-AC03-76SFOOO98.

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Shapes and sizes of nanoscale Pb inclusions in Al