Pb nanoprecipitates in Al: Magic-shape effects due to elastic strain

Transcription

1 Downloaded from orbit.dtu.dk on: Nov 04, 018 nanoreciitates in Al: Magic-shae effects due to elastic strain Hamilton, J.C.; Leoard, F.; Johnson, Erik; Dahmen, U. Published in: Physical Review Letters Link to article, DOI: /PhysRevLett Publication date: 007 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Hamilton, J. C., Leoard, F., Johnson, E., & Dahmen, U. (007). nanoreciitates in Al: Magic-shae effects due to elastic strain. Physical Review Letters, 98(3), DOI: /PhysRevLett General rights Coyright and moral rights for the ublications made accessible in the ublic ortal are retained by the authors and/or other coyright owners and it is a condition of accessing ublications that users recognise and abide by the legal requirements associated with these rights. Users may download and rint one coy of any ublication from the ublic ortal for the urose of rivate study or research. You may not further distribute the material or use it for any rofit-making activity or commercial gain You may freely distribute the URL identifying the ublication in the ublic ortal If you believe that this document breaches coyright lease contact us roviding details, and we will remove access to the work immediately and investigate your claim.

2 Nanoreciitates in Al: Magic-Shae Effects due to Elastic Strain J. C. Hamilton, 1 F. Léonard, 1 E. Johnson, and U. Dahmen 3 1 Sandia National Laboratories, Livermore, California, USA Nano Science Center, Niels Bohr Institute, University of Coenhagen, Coenhagen, Denmark and Deartment of Materials Research, Riso National Laboratory, Roskilde, Denmark 3 National Center for Electron Microscoy, Lawrence Berkeley National Laboratory, Berkeley, California, USA (Received 7 November 006; ublished 5 June 007) We resent a theory for size-deendent shaes of nanoreciitates in Al, introducing the concet of magic shaes, i.e., shaes having near-zero homogeneous elastic strains. Our quantitative atomistic calculations of edge energies show their effect on reciitate shae to be negligible, thus it aears that shaes must be due to the combined effect of strain and interface energies. By emloying an algorithm for generating magic shaes, we relicate the exerimental observations by selecting magic-shae reciitates with interfacial energies less than a cutoff value. DOI: /PhysRevLett A major goal of nanoscience is to understand and control the roerties of functional nanostructures, including, for examle, catalyst articles, quantum dots on surfaces, and inclusions in alloys. Such roerties are often determined by the nanostructure shae. The Wulff construction, based on minimization of interfacial energy subject to a constraint of constant volume, redicts that the cluster shae will be indeendent of cluster size. Exerimentally, however, changes in cluster shaes are often observed as cluster sizes aroach the nanoscale. Conventionally such sizedeendent shae effects are attributed to the increasing contribution of edge energies relative to interfacial energies at small size. For examle, edge energies are included in theories for the shaes of snow crystals [1], discussions of surface faceting [], theories for the shaes of strained Ge yramids grown on a surface [3], and discussions of inclusions in bulk Al [4]. Until very recently [5,6], no atomistic calculations of edge energies had been ublished. Here we resent a theoretical examination of sizedeendent shaes observed for nanoreciitates in Al [4,7]. Using the embedded atom method (EAM) with otentials from Landa and co-workers [8], quantitative calculations of the total reciitate energies and relevant edge energies were erformed. We find the effect of edge energies on reciitate shae to be negligible for all sizes and conclude that observed size-deendent shae effects must be exlained by the minimization of interfacial energy and strain energy. We resent an algorithm for generating reciitate shaes with very small homogeneous strain energies and use the term magic shaes to describe these secial reciitate shaes. The minimization of reciitate interfacial energies subject to the magicshae constraint accounts for the observed size-deendent shae effects. Figure 1 summarizes the exerimental TEM observations for reciitates formed by ion imlantation and annealing [7]. A few of the smallest reciitates are octahedral, bounded by f111g-tye =Al interfaces. The largest reciitates are aroximately tetrakaidecahedral, PACS numbers: , Bc, 61.7.Qq, 6.0.Dc bounded by f111g- and f100g-tye =Al interfaces (see uer right inset of Fig. 1). The lotted asect ratio is C= A C= A 1 A where C is the sacing between a air of f100g-tye interfaces, and A 1 and A are the sacings between the two airs of f111g-tye interfaces. EAM calculations for =Al interfacial free energies have reviously been reorted [9] giving :44 mev= A and :44 mev= A at T 400 K (in the vicinity of the exerimental annealing temerature). The redicted asect ratio at 400 K from the Wulff construction is C=A 100 =g 111 1:165. For larger reciitates, the measured asect ratio is close to this value; however, as seen in Fig. 1, smaller reciitates behave quite differently, and exhibit a wide range of asect ratios. To understand these shae effects, we calculated reciitate energies for octahedral, tetrakaidecahedral, and cuboctahedral reciitates inserted into similarly shaed voids in a block of Al with eriodic boundary conditions. The sizes and shaes of the reciitates and the Al voids were selected to obtain relatively small homogeneous elastic strains. Figure comares the calculated reciitate energies for the different shaes over a range of reciitate sizes. The x axis is N 1=3 which, for a given shae, is roortional to the edge length, thus allowing comarison of energies for reciitates with similar volumes (i.e., similar N 1=3 ). For this 0 K calculation, the Wulff construction redicts C=A 1:31, based on T 0KEAM interface energies, :38 mev= A and 111 7:76 mev= A. However, our EAM calculations show that these three shaes are cometitive in energy over various arts of this size range. In order to understand the roles of interface, strain, and edge energies in determining reciitate shaes, it is necessary to searate their resective contributions to the reciitate energy, E reci E inter E strain E edge E vertex. To be secific, we illustrate by resenting the calculations of the edge energy " for octahedral reciitates inserted in octahedral Al voids =07=98(3)=3610(4) The American Physical Society

3 FIG. 1 (color). The uer right-hand inset shows a tyical reciitate in Al with the reciitate size defined as A A 1 A =. The red oints show exerimentally measured asect ratios for a large exerimental samle of such nanoreciitates. The blue line shows the asect ratio redicted by edge energies from the quantitative atomistic calculations described in the text. The square symbols lot the redictions of the magic-shae theory described in the text. The uer left inset shows two reciitates with a notch along one edge, also consistent with the magic-shae theory. We started with a block of aroximately 10 5 Al atoms with eriodic boundary conditions. Next an octahedron of Al atoms with n oct atoms on each edge was removed. This void was filled by inserting an octahedron of atoms with m oct atoms on each edge. Because the lattice constants of and Al are in the ratio a =a Al 11=9, only very small, elastic strains are generated if n oct =m oct 11=9. The starting configurations are naturally categorized by the value of n oct m oct as shown in Table I.Foragiven, a small range of values for m oct will corresond to reciitates with relatively small homogeneous strain. The total EAM energy E total was determined by relaxing atomic ositions using conjugate gradient energy minimization. From E total, the reciitate energy was obtained by subtracting the total cohesive energies, thus E reci E total N Al E Al N E, where N is the number of atoms in the reciitate, N Al is the number of Al atoms in the Al matrix surrounding the reciitate, and E Al and E are the bulk cohesive energies. The total number of atoms in the eriodic simulation cell is N Al N. E reci is lotted by the red oints and fitted curves in the uer ortion of Fig FIG. (color). Summary of EAM calculations for in Al. The uer lot shows E reci as a function of the cube root of the number of atoms in the reciitate. The insets show the three shaes considered: octahedra, tetrakaidecahedra, and cuboctahedra. The relative reciitate energies for these three shaes can be comared at a articular value of N 1=3 (corresonding to a given volume). The lower lot is constructed from the octahedra calculations by subtracting E inter A from the reciitate energy, leaving E strain E edge E vertex (lotted as red oints) and fit by a arabola for each value of (see Table I). The minima of the arabolas reresent the energies of zero-strain octahedral reciitates. The sloe of the black line tangent to the arabolas gives the edge energy " :5 mev= A. The next ste will be to calculate E reci E inter E strain E edge E vertex. Before discussing the details of this calculation, we wish to oint out that for the lowest strain reciitates, E reci is less than E inter, and thus E strain E edge E vertex is negative (as shown in the lower ortion of Fig. ). Since the strain energy can only be zero or ositive, this imlies that E edge E vertex is negative. The quantitative calculation of these quantities follows. A revious ublication [5] has described the calculation of edge energies for free Pd nanoclusters. A recise definition of edge length, based on the concet of the Gibbs equimolar dividing surfaces, was found to be absolutely essential in order to define and quantify the edge and vertex energies [10]. We use that definition to write r s oct m 3 oct m 3 oct a : Here s oct is the edge length of the octahedron, m oct is the

4 TABLE I. This table lists some of the octahedral reciitate configurations used for the EAM calculations. These configurations have relatively small homogeneous strain and corresond to the lowest strain oints lotted in red in Fig.. Here, n oct is the number of Al atoms on an edge of the removed Al octahedron and m oct is the number of atoms on an edge of the inserted octahedron. These cases are selected to have small homogeneous elastic strains by choosing n oct =m oct a =a Al, and are groued according to the value of n oct m oct m oct 3, n oct 4 m oct 7, n oct 9 m oct 1, n oct 15 m oct 17, n oct 1 m oct 4, n oct 5 m oct 8, n oct 10 m oct 13, n oct 16 m oct 18, n oct m oct 5, n oct 6 m oct 9, n oct 11 m oct 14, n oct 17 m oct 19, n oct 3 m oct 10, n oct 1 m oct 15, n oct 18 number of atoms on an edge (including a vertex atom at each end), and a is the lattice constant of lead. The total area of an octahedron is A s oct. By subtracting the total interface energy, we find E reci A E strain E edge E vertex. This quantity is lotted in red in the lower ortion of Fig. for the, n oct, and m oct values listed in Table I. Points for octahedra having 5 or 6 are also lotted. These oints are fit by arabolas since the strain energy for small homogeneous strains can be written as E strain s oct s o where s o reresents the octahedron edge length corresonding to zero elastic strain for a given value of, and is a constant for a given value of. The strain energy can be estimated from analytical models for a sherical inclusion in a sherical void [7]. Curvatures of the arabolas lotting E strain are in good agreement with that estimate. To extract the edge energy, a tangent line was constructed to the arabolas having, 3, 4, 5, and 6, as lotted in black in the lower section of Fig.. The sloe of that tangent line gives the edge energy. This tangent line misses the minimum of the 1 arabola, suggesting that the searation into edge and vertex energies is roblematic for extremely small octahedra with 3, 4, or 5 atoms on an edge. The edge energy calculated from the sloe of the tangent line is " :5 mev= A. In order to calculate " we used the same rocedure, but considered only cuboctahedral reciitates. For cuboctahedra, a different formula (see Ref. [5]) must be used to relate the edge length of a cuboctahedra s cub to m cub, the number of atoms on an edge. The calculated edge energy is " : mev= A. The two edge energies for reciitates in Al are essentially equal. We next consider the ossible role of edge energy and interface energy based on the assumtion that residual strains may be neglected. This is consistent with the observation that inclusions have nearly zero residual strain when they adot magic-size dimensions [4]. For this case we write the energy of a truncated octahedral reciitate in the form E E inter E edge. To find the minimum energy for constant volume V, as a function of shae, we use Lagrange multiliers to minimize the function, F E= 111 V 3 s 6 3 t 1 s s 3 =3 t 3 ; where s is the edge length of the untruncated octahedron, t is the truncation length, 100 = 111, "= 111, and " " " :5 mev= 0 and relacing the arameters t and s with aroriate combinations of the measured arameters C and A, we obtain the asect ratio as a function of size A as follows: C= A 6 = A = 1 = A. This function (calculated using 400 K interfacial free energies) is lotted in Fig. 1 as a blue line. It is aarent from the lot that these edge energies have essentially no effect on the shae of nanoreciitates in Al. We have also confirmed that uncertainties in the edge energies on the order of " " mev= A will change the asect ratio by only a very small amount. We also note that edge energy cannot exlain the increasing scatter in asect ratio observed for the smaller reciitates. Since edge energy cannot be resonsible for these effects, and they cannot be exlained solely by interface energies (as in the Wulff construction), we turn our attention to strain energy, the only remaining ossibility. In the following we consider how to construct a set of ossible reciitate shaes having zero (or very small) homogeneous strain. We start by observing that an fcc lattice can be built using two fundamental building blocks, a square yramid (with one atom at each vertex) and a tetrahedron (also with one atom at each vertex). The rhombohedral rimitive unit cell of the fcc lattice has one octahedral interstitial site and two tetrahedral interstitial sites, and can be constructed by lacing two square yramids base to base, forming an octahedron, and adding two tetrahedra laced on oosing triangular faces of this octahedron. If one wants to build strain-free fcc reciitate nanoclusters in Al, it can only be done by assembling tetrahedral and square yramid building blocks having 9 atoms on an edge. Such nanoclusters can be laced in a void created by removing the same Al shae having 11 atoms on an edge. In order to build these shaes with zero strain, the edge length of the tetrahedra and the square yramids must be 9 times the nearest neighbor distance or 31.5 Å. The concet of magic sizes, discussed in the literature [7], can be relaced by the criteria of magic shaes built from tetrahedra and square yramids with edge length s bb 31:5 A. For any shae constructed of these building blocks, the strain energy is

5 zero, and the interfacial energy can be calculated based on the interfacial areas, A 111 and A 100, of the shae assembled from these building blocks. In reality, we are interested not only in comletely strain-free reciitates, but also in reciitates having very small homogeneous strains. Examles of such reciitates are listed in Table I in the columns labeled 1 and 3. To cover cases having odd as well as even integer values of requires working with smaller structural building blocks having edge length s bb 15:75 A. (The shaes with zero strain discussed in the revious aragrah can of course be assembled from these smaller building blocks.) The magic shaes are the set of ossible reciitate shaes (and sizes) with zero or small strain, built with these building blocks with edge length s bb 15:75 A. We now have in hand the tools needed to exlain the exerimental data. In articular we have an algorithm to generate a set of reciitates with zero or small homogeneous strain. To illustrate, we consider reciitates with O h symmetry, with shaes generated by starting with an octahedron with edge length s, and removing 6 square yramids with edge length t. In order that the reciitates be nearly strain free, the concet of magic shaes requires that s must be quantized in units of s bb 15:75 A, i.e., s s bb where is a ositive integer. Similarly, t must be quantized as t qs bb, where q is an integer 0 q<=. Since these reciitates are nearly strain free, and since we have shown that edge energies make only a negligible contribution to shae effects, the reciitate energies are aroximately equal to the interface energies, E reci E inter A A The asect ratio is C=A 1 t=s 3 3, the total (100) area is A100 6t, and the total (111) area is A 111 8s 4t. Here we use the T 400 K interfacial free energies in order to redict the C=A ratio. At this oint, we know the asect ratio and the interface energy of all the (nearly) stain-free reciitates with O h symmetry. In order to redict the observed asect ratios as a function of reciitate size, we need a criterion for deciding which reciitate energies might actually occur in a quasiequilibrium distribution of reciitates. For a given reciitate volume, it is easy to calculate the interfacial energy E Wulff of a reciitate with the Wulff shae. Since this is the lowest ossible energy for a given volume, we find it convenient to work with the energy E E reci E Wulff. For reciitates at equilibrium, one would exect a Boltzmann distribution of reciitate shaes with a characteristic e E=kT robability. In this exeriment, the reciitates are not at equilibrium and Ostwald riening would continue slowly during further annealing. In order to reroduce the exerimental shae distribution (shown as red oints in Fig. 1), we rejected all reciitate sizes and shaes having E>60 ev. While not rigorous, it aears that this energy criterion allows a shae distribution which, while far from equilibrium, is slowly riening at the annealing temerature thereby avoiding shaes with very large E. In Fig. 1 the black symbols reresent the comlete set of reciitate articles meeting four conditions: O h symmetry, magic shae (i.e., near zero strain), E 60 ev, and A 55 A. The agreement between this magicshae theory and the exerimental data is seen to be good. Finally we note that, in addition to the reciitates with O h symmetry, there are many other reciitates which break this symmetry. In all cases, the exerimentally observed shaes are consistent with the concet of magic shaes and nearly zero strain. The insets in Fig. 1 show two such cases where the symmetry is broken. In the uer right-hand inset the symmetry is broken because A 1 A, while in the other inset the symmetry is broken by removing a row of tetrahedra and square yramids to form a notch at an edge along the viewing direction. To summarize, the model resented in this work accounts for the observed behavior of nanoscale solid inclusions. In articular, it is shown that while edge energy has a negligible effect on shae, strain energy leads to a sequence of magic shaes. The increasing granularity of such magic shaes at smaller sizes exlains the increased scatter around the Wulff shae observed exerimentally. We exlain the exerimental data by assembling magic shaes from building blocks which are tetrahedra and square yramids with edge lengths s bb 15:75 A. Given a set of magic shaes, we find that the members of this set with relatively small interface energies are the shaes that are observed. By roviding an exlanation for all the exerimental data, our model offers an understanding of how nanoscale inclusions aroach equilibrium under constraint within a solid matrix. This work was suorted by the U.S. Deartment of Energy, Basic Energy Sciences, Division of Materials Science, under Contract No. DE-AC04-94AL Additional suort was received from the Danish Natural Sciences Research Council. [1] J. Nelson, Philos. Mag. A 81, 337 (001). [] C. H. Nien and T. E. Mady, Surf. Sci. 380, L57 (1997). [3] O. E. Shklyaev, M. J. Beck, M. Asta, M. J. Miksis, and P. W. Voorhees, Phys. Rev. Lett. 94, (005). [4] E. Johnson, A. Johansen, U. Dahmen, S. Chen, and T. Fujii, Mater. Sci. Eng. A , 187 (001). [5] J. C. Hamilton, Phys. Rev. B 73, (006). [6] C. M. Retford, M. Asta, M. J. Miksis, P. W. Voorhees, and E. B. Webb, Phys. Rev. B 75, (007). [7] U. Dahmen, S. Q. Xiao, S. Paciornik, E. Johnson, and A. Johansen, Phys. Rev. Lett. 78, 471 (1997). [8] A. Landa, P. Wynblatt, D. J. Siegel, J. B. Adams, O. N. Mryasov, and X. Y. Liu, Acta Mater. 48, 1753 (000). [9] A. Landa, P. Wynblatt, E. Johnson, and U. Dahmen Acta Mater. 48, 557 (000). [10] The requirement for recise definition of interface surfaces was mentioned briefly in C. Rottman and M. Wortis, Phys. Re. 103, 59 (1984)