Generalized Nearest-Neighbor Broken-Bond Analysis of Randomly Oriented Coherent Interfaces in...

Generalized Nearest-Neighbor Broken-Bond Analysisof Randomly Oriented Coherent Interfacesin Multicomponent Fcc and Bcc Structures

B. SONDEREGGER and E. KOZESCHNIK

In this article, a model for the estimation of matrix/precipitate interfacial energies is developed.The classic nearest-neighbor broken-bond (NNBB) model is taken as a basis and furtherdeveloped, to (1) take into account atomic bindings over an arbitrary number of neighboringshells and (2) account for general, multicomponent solid solutions. The model is sufficientlysimple and yet reliable for providing estimates of interfacial energies in applications in complex,time-consuming computer simulations of a microstructure/precipitate evolution in which moresophisticated approaches cannot be used. As an example, the model is applied to randomlyoriented interfaces in fcc and bcc crystal structures. It is shown that both kinds of crystalinterfaces, fcc and bcc, exhibit roughly the same mean interfacial energies, as long as a sufficientnumber of nearest-neighbor shells is taken into account. A comparison with published experi-mental and theoretical data on interfacial energies shows good agreement.

DOI: 10.1007/s11661-008-9752-6� The Minerals, Metals & Materials Society and ASM International 2009

I. INTRODUCTION

INTERFACIAL energies play an important role inthe nucleation, growth, and coarsening of precipitatesand crystal grains. Unfortunately, a sufficient quantity isnot available for direct experimental measurement andmost of the published data on interfacial energies havebeen derived from indirect methods. Most prominent isthe fitting of the experimental results on nucleation[1–5]

or particle coarsening[6–10] to the corresponding kineticmodels.

The lack of direct experimental data leads to anumber of different numerical methods for modelinginterfacial energies. In recent years, growing computercapacities opened the door to increasingly sophisticatedapproaches. Price and Cooper,[11] Rapcewicz et al.,[12]

Sluiter et al.,[13,14] Hartford,[15] and Benedek et al.[16]

applied first-principle studies, solving the many-bodySchrodinger equations. Landa et al.[17] and Wynblatt[18]

are using the ‘‘glue potential method’’[19] and combin-ing it with Monte Carlo simulations. Staron andKampmann[20] apply a cluster dynamic approach andAsta et al.[21–23] apply ab initio and cluster expansionmethods, in order to calculate interfacial energies.

Mishin[24] and others use the embedded atom method(EAM).Whereas these models can predict interfacial energies

with good accuracy, they are not practical for imple-mentation with complex simulation algorithms, forexample, of microstructure/precipitate evolution,because of computational costs that are too high.Simulation tools, such as MatCalc[25–27] or otherKampmann Wagner–type models,[3] e.g., by Robsonet al.,[4,5,28] require fast estimations of interfacialenergies for various phases in a given chemical environ-ment. For this task, sufficiently simple and accurateexpressions are needed. The classic nearest-neighborbroken-bond (NNBB) concept meets these demands,because the interfacial energy is expressed as a single,closed equation. The only input data (enthalpy ofsolution) are available from CALPHAD-type thermo-dynamic databases.[29] Therefore, in the present work,this approach is taken as the conceptual basis for thedevelopment of an extended model that takes intoaccount all neighbor interactions as well as generalmulticomponent solutions. Furthermore, this articledoes not focus on finding appropriate atomic potentialsin various materials, but applies Lennard-Jones–typepotentials as well as EAM potentials taken from theliterature. It is shown that the dependence of the meaninterfacial energy on the chosen potentials is ratherweak, in most cases.The grounds for the NNBB model were laid by Bragg

and Williams.[30] These authors assumed that the bind-ing energy of a crystal can be expressed as the sum of thebinding energies of the neighboring atoms. This conceptwas used by Becker in his treatment of interfacialenergies.[31] Later, Turnbull[32] suggested that the sum ofall individual nearest-neighbor atomic binding energies

B. SONDEREGGER, Assistant Professor, is with the Institute forMaterials Science, Welding, and Forming, Graz University ofTechnology, A-8010 Graz, Austria, and the Materials Center LeobenForschung GmbH, A-8700 Leoben, Austria. Contact e-mail: [email protected] E. KOZESCHNIK, University Professor,formerly with the Institute for Materials Science, Welding, andForming, Graz University of Technology, is with the ChristianDoppler Laboratory for ‘‘Early Stages of Precipitation,’’ Institute forMaterials Science and Technology, Vienna, and the Institute forMaterials Science and Technology, Favoritenstrabe 9-11, 1040 Vienna,Austria.

Manuscript submitted November 10, 2007.Article published online January 21, 2009

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MARCH 2009—499

be replaced by a single thermodynamic quantity, i.e., theenthalpy of solution.

Turnbull has developed his concept on the example of[111]-oriented planes in an fcc structure, assuming first-nearest-neighbor interactions. Over the decades, theNNBB approach has been extended in several direc-tions, i.e., by taking into account diffuse interface layers,orientational dependencies, and different crystal systems(simple cubic, fcc, bcc, …) or by considering not onlynext-nearest, but also second-, third-, … nearest neigh-bors. These approaches will be briefly reviewed next.

In the classic NNBB model, the assumption is madethat the thickness of the interface separating the twocrystals is infinitely small. This so-called ‘‘sharp inter-face’’ approximation is valid, however, only within thelimit of sufficiently low temperatures. At higher temper-atures, and particularly when approaching the solubilitylimit of the precipitates, the interface usually becomesdiffuse. Several authors have, therefore, introduceddiffuse interface models to take into account theentropic contributions of interfaces with a finite thick-ness. Guggenheim[33] and Prigogine and Defay[34] madecalculations based on a single intermediate layer. Ono[35]

was the first to define a model in which the thickness ofthe interface evolves as a consequence of the theoreticalmodel and need not be defined a priori. Independently,Hillert developed an equivalent approach some yearslater.[36,37] However, he did not apply his model tointerfaces, but to periodic concentration variations inthe treatment of spinodal decomposition.

All the models mentioned here define coupled non-linear differential equations that cannot easily be solvedanalytically and that thus require extensive numericaltreatment. Motivated by these difficulties, Cahn andHilliard[38] translated the lattice-based description ofinterfaces into a continuums-mechanical framework.Here, the mathematical treatment became somewhateasier; nevertheless, the resulting integrals still requiresolution by numerical methods. An important leapforward in the treatment of interfaces was achieved byKahlweit;[39] after linearizing the concentration profilesat the interface, an approximate analytical solution forthe energy of interfaces with a finite thickness wasobtained.

When treating interfaces in the framework of lattice-based models, an important issue is the anisotropy ofthe interfacial energy with orientation. Meijering[40]

computed and compared the interfacial energies ofdiscrete lattice planes with low index numbers based onthe NNBB model. Nicholas[41] evaluated the orientationdependence of surface energies in different crystalsystems for up to 60 different orientations. In afundamental article of Lee and Aaronson,[42] the taskof considering orientation is solved comprehensively,allowing arbitrary orientations of the interface. BothMeijering and Lee and Aaronson find only a weakorientation dependence of the interfacial energy, with amaximum spread of 30 pct. Later, Yang and Enomot-o[43,44] applied the methodology of Lee and Aaronson toother crystal systems.

Although the majority of treatments of interfacialanisotropy have been derived for general crystal

structures, application was demonstrated mainly on fccsystems (Turnbull,[32] Ono,[35] Meijering,[40] and Lee andAaronson[42]). Nicholas[41] also considered the bcclattice; however, he only evaluated surface energies,i.e., the energies of free crystal surfaces. Yang andEnomoto[43,44] investigated the interface between B1-type precipitates (fcc and bcc NaCl-structure) in aus-tenite and ferrite.An important step in advancing the NNBB model has

been the extension from first-nearest to second-, third-,… nearest neighbors. In calculating the energy of freesurfaces, Nicholas[41] considered all pairwise combina-tions of bonds between the atoms, independent of theirdistance. Still, the initial concept is conserved and thecrystal energy is calculated from the superposition ofpairwise interaction potentials. A major result of thework of Nicholas is that significantly improved esti-mates of interfacial energies are already achieved, ifsecond-nearest neighbors are also considered. Lee andAaronson[42] also took into account more distantneighbor shells; however, they limited their treatmentto first-nearest-neighbor interactions in specific atomicplanes, a method also adopted by Yang and Enomot-o.[43,44] The latter combined this concept with the resultsof Miedema,[45] who determined the interaction energiesof atoms in a semi-empirical way. Table I gives anoverview of various approaches in extension of theoriginal NNBB concept.In Section II, the extension of the NNBB model to a

generalized broken-bond (GBB) concept is developed.The interfacial energies and their anisotropies will beevaluated and discussed in reference to the example ofrandomly oriented coherent interfaces in fcc and bcclattices.

II. CALCULATION OF INTERFACIALENERGIES: GBB CONCEPT

When extending the NNBB concept, the originalresults of Becker[31] and Turnbull[32] are taken as astarting point. Both authors consider a phase a consist-ing of atoms A, a phase b consisting of atoms B, and asharp interface between the two phases. The energy ofsolution DEsol and the interfacial energy c can then bewritten as the sum of the binding energies of theneighboring atoms with

DEsol ¼ N�zL�e� ½1�

c ¼ nS�zS�e� ½2�

where

e� ¼ eBA �1

2eAA þ eBBð Þ ½3�

The term zS denotes the number of broken bonds perinterface atom, zL is the total number of bonds peratom, and nS is the number of atoms per unit interfacearea. The terms eAA, eBB, and eBA represent pairwisebond energies between atoms A and B.

500—VOLUME 40A, MARCH 2009 METALLURGICAL AND MATERIALS TRANSACTIONS A

In References 32 and 38 through 44, e* is applied totwo- or three-component systems with atoms A and Band, eventually, C. However, analogous expressions fore* can also be derived for multicomponent systems, asshown in Appendix I. Thus, the following consider-ations are valid for systems with an arbitrary number ofcomponents.

In the original treatment, zS, zL, and e* are defined forfirst-nearest-neighbor interactions. In order to addition-ally include second, third, and …kth-nearest neighbors,Eqs. [1] and [2] must be generalized with the goal offinding the ‘‘effective’’ structural parameters zS,eff andzL,eff instead of zL and zS. For this reason, a weightingfunction is introduced, to account for the decreasingbinding energies to more distant atoms, counted relativeto the nearest-neighbor bond. In the following calcula-tions, this weighting function is denoted as f(rk/r1), withr1 the distance between two first-nearest neighbors andrk the distance to the kth-nearest neighbor (or kth‘‘shell’’). With e*,k denoting the binding energy to thekth-nearest neighbor (e* is now written e*,1), the bindingenergy to a kth-nearest neighbor can be written as

e�;k ¼ e�;1�f rk=r1ð Þ ½4�

The summation of all shells and the rearrangement ofterms delivers

DEsol ¼ N�e�;1�zL;eff ½5�

c ¼ nS�e�;1�zS;eff ½6�

with

zL;eff ¼X1

k¼1zL;k�f rk=r1ð Þ ½7�

zS;eff ¼X1

k¼1zS;k�f rk=r1ð Þ ½8�

where zL,k denotes the coordination number of thekth shell and is given by the crystal lattice structure

and zS,k represents the number of broken bondsin the kth shell. This expression is more involvedthan the expression for zL,k and is discussed in detailin the Section III. A combination of Eqs. [5] and [6]finally leads to the desired result for the interfacialenergy:

c ¼ nS�zS;effN�zL;eff

�DEsol ½9�

The major task remaining is to find values for nS, zL,eff,and zS,eff, for specific crystal systems and interfaceorientations. This is next demonstrated for interfaces inthe fcc and bcc crystal system.The number of atoms at the interface nS in Eq. [9] can

be straightforwardly defined in terms of the volumetricpacking density nV (number of atoms per cubic meter)with

nS ¼ n2=3V ½10�

With this representation, nS is not orientation depen-dent and calculates as nS;fcc ¼ 42=3=a2 for fcc andnS;bcc ¼ 22=3=a2 for bcc lattices, with a being the latticeconstant. In the present approach, because this defini-tion of nS excludes the interface orientation (as doesthe effective coordination number zL,eff), any effects ofthe anisotropy of the interfacial energy must beincorporated into the effective number of brokenbonds across the interface zS,eff, counted per interfaceatom.For a general analysis of the number of broken bonds

across an interface zS,eff of arbitrary orientation, thefollowing steps must be carried out.

(a) Define the crystal structure and the orientation ofthe interface. Let the interface divide the latticeinto two regions, a and b.

(b) Count the effective number of broken bonds acrossthe interface area A. The weighting factor for eachbond is given by f(rk/r1).

(c) Normalize the number of broken bonds withrespect to the mean surface area Afcc = a2/42/3 andAbcc = a2/22/3 of one interface atom.

Table I. Extensions to Original NNBB Concept

Author(s) Year

DiffuseInterfaceLayer

OrientationDependence

CrystalSystem Multicomponent

nth NearestNeighbors

Interface/Surface Reference

Becker 1938 — — simple cubic no — I 31Guggenheim 1945 yes — — no — S 33Ono 1947 yes — fcc no — I 35Prigogine et al. 1950 yes — — no — S 34Turnbull 1955 — — fcc no — I 32Cahn et al. 1958 yes — liquid no — I 38Kahlweit 1962 yes — liquid no — I 39Meijering 1966 yes yes fcc no — I 40Nicholas 1968 — yes fcc, bcc no yes S 41Lee et al. 1980 yes yes fcc no yes I 42Yang et al. 2001 yes — B1 in fcc three-component yes I 43Yang et al. 2002 yes yes B1 in bcc three-component yes I 44Sondereggerand Kozeschnik

2009 — yes fcc, bcc multicomponent yes I this article


For the last step, the total number of effective brokenbonds across the interface beff must be divided by thenumber of interface atoms (AÆnS), delivering

zS;eff ¼beffA�nS

½11�

Figure 1 shows a two-dimensional (2-D), representativeexample of possible broken bonds in a simple cubiclattice. The basic idea of bond counting is to assign allthe broken bonds of atoms in phase a to the interfacearea A, where the normal projection of the atomposition falls into the area A; if the normal projectionfalls into a neighboring region A¢, the bonds are assignedto the area A¢, etc. The position of the correspondingatom in phase b is of no importance. By this technique,one gets an unambiguous assignment of broken bondsand interface areas. In Figure 1, the atoms in phase a aremarked black; the corresponding volume in phase a ismarked gray.

In analyzing the different types of bonds shown inFigure 1, bond 1 represents the simplest variant, withlength r1 directly across the interface. Bond 2 is a higher-order nearest-neighbor bond. This bond is not countedin the original NNBB model. The bond is taken intoaccount here, because the projection of the a atom fallsinto the interface region A. Bond 3 is also considered asbelonging to the interface region A. According to themodel of Lee and Aaronson,[42] this bond would havebeen neglected (they only count the nearest neighbors ineach atomic plane parallel to the interface). In contrastto bond 3, bond 4 is not counted in the present model,because the normal projection of the a atom falls outsidethe interface region A.

In the practical numerical analysis, periodic volumeelements are used. With these volume elements, theentire lattice can be reconstructed by periodic transla-tional reproduction. Figure 2 shows an example of a

2-D element for a [3,1] interface. The periodic volumeelement is located in phase a and, usually, only a fewatoms are needed for the construction. The width of thevolume element is given by the [3,1] lattice vector, thethickness by the highest considered distance rk. Allatoms in a up to this distance rk can have broken bondswith atoms in b. In b, all atoms have to be consideredwith a distance smaller than rk (Figure 2). Note that twolinearly independent lattice vectors with integer indicesmust be defined for three-dimensional interfaces.If the numerical analysis is performed to a given shell

index k, interactions from all atoms with a higher k areneglected. However, the influence of these bonds canconveniently be taken into account with an analyticalcorrection term (more details are available in AppendixII). Assume that all atoms with a shell index larger thank are homogeneously distributed in the crystal volume.The influence of all outer shells can then be incorporatedinto correction terms DzL,corr and DzS,corr, which areevaluated in analogy to Eqs. [7] and [8] with

DzL;corr ¼Z1

r¼rk

nV�f rk=r1ð Þ�dV ½12�

The symbols used in Eq. [12] and a solution for theintegral are given in Appendix II. When assuming, forexample, that the interatomic potentials decrease withr�6, the integration of Eq. [12] delivers

DzS;corr ¼ 2p�n4=3V �r61�Z1

aþb>r100

aþ bð Þ�5b�da�db ½13�

and

DzL;corr ¼ nV�4p3r61�r�3k ½14�

DzS;corr ¼p2�n4=3V �r61�r�2k ½15�

Fig. 1—Broken bonds across a randomly oriented interface.Fig. 2—Schematic of a periodic volume element for reconstructionof infinite 2-D interfaces with [3,1] orientation.


It must be noted that the choice of interactionpotential with r�6 (the Lennard-Jones potential[46]) issomewhat arbitrary; however, if an exact solution isdesired, it must be performed in every system underconsideration. This is generally a complex task. To keepthe present treatment simple, interfacial energies areevaluated only for examples of a few model potentials(Tables II and V). For these EAM potentials, thecalculation for the correction term is not carried out;however, the deviation relative to the exact solution issmaller than 1 pct, as shown later in Figure 4.

It is also important to emphasize that the analyticalcorrection terms are isotropic and, thus, any anisotropyof the crystal lattice (e.g., from cubic symmetry) areneglected. Nevertheless, consideration of these terms inthe evaluation of interfacial energies significantly im-proves the accuracy of the analysis. This is demon-strated in Section III.

III. ANALYSIS OF INTERFACIAL ENERGIES

In this section, a quantitative evaluation of theeffective coordination number zL,eff and the effectivenumber of broken bonds zS,eff is presented. First, zL,eff isanalyzed in detail; next, the interfacial energy structuralfactor zS,eff/zL,eff is examined. In the numerical analysis,interface orientations with integer indices up to 10 areinvestigated; higher-order interfaces between the [a,b,c]directions are interpolated.

For the evaluation of zL,eff, an ideal r�6 dependence ofthe interaction potential between the atoms is firstassumed. This functional dependency corresponds to theclassic Lennard-Jones potential for free atoms.[46] Later,

values for r�7 and r�8 and tight-binding EAM potentialsare also presented for comparison. With Eqs. [7] and[14], the effective coordination numbers for fcc and bcccrystal structures are calculated with zL,eff,bcc = 12,234and zL,eff,fcc = 14,442. These values differ significantlyfrom the traditional first-nearest-neighbor values ofzL,bcc = 8 and zL,fcc = 12.Clearly, taking into account a lower number of

nearest-neighbor shells will deliver less accurate resultsand, generally, the effective coordination numbers areunderestimated, i.e., the values are too low. Figure 3shows the deviation of the effective coordination num-bers zL,k,eff when only considering k shells, compared toa reference value obtained for k = 500. One curverepresents the deviation without analytical correction;the other includes this correction (Eqs. [14] and [15]).Figure 3 shows that, if only first-nearest neighbors are

considered in the analysis, the binding energies aresignificantly underestimated (by 53 pct in bcc and 20 pctin fcc) compared to the value for 500 nearest-neighborshells. Even with second-nearest neighbors, the devia-tion from the values of the full analysis remains wellover 10 pct. If maximum deviations below 1 pct aredesired, it is necessary to include bond interactions up tothe 12th-nearest neighbor, which corresponds toapproximately 200 atoms. The analytical correctionimproves the result significantly: for fcc, only five shellsare necessary to reach an accuracy of 1 pct; for bcc, sixshells are sufficient.Similar results are obtained for the structural factor

zS,eff/zL,eff: the higher the number of participating shellsk, the more accurate the results. Figure 4 shows thedeviation of the lower-index effective structural factorrelative to the benchmark of 500 shells, including the

Table II. The zS,eff/zL,eff of Selected Low-Index Boundaries, Assuming r26, r27, and r28 Interatomic Potentials

Direction

r�6 r�7 r�8

Fcc Bcc Fcc Bcc Fcc Bcc

[1,0,0] 0.3105 0.3332 0.2848 0.3113 0.2746 0.3041[1,1,0] 0.3271 0.2905 0.3019 0.2617 0.2918 0.2488[1,1,1] 0.2871 0.3351 0.2581 0.3096 0.2454 0.2986[2,1,0] 0.3394 0.3283 0.3152 0.3029 0.3056 0.2925[2,1,1] 0.3222 0.3224 0.2953 0.2956 0.2839 0.2837[2,2,1] 0.3182 0.3281 0.2907 0.3012 0.2790 0.2893[3,1,0] 0.3372 0.3355 0.3126 0.3116 0.3028 0.3025[3,2,0] 0.3384 0.3174 0.3138 0.2904 0.3039 0.2790[3,1,1] 0.3275 0.3361 0.3019 0.3111 0.2913 0.3007[3,2,1] 0.3322 0.3198 0.3063 0.2921 0.2955 0.2796[3,2,2] 0.3118 0.3319 0.2838 0.3056 0.2717 0.2939[3,3,1] 0.3244 0.3198 0.2977 0.2921 0.2866 0.2798[3,3,2] 0.3094 0.3328 0.2812 0.3066 0.2691 0.2951[4,1,0] 0.3337 0.3379 0.3088 0.3144 0.2988 0.3057[4,3,0] 0.3365 0.3110 0.3116 0.2835 0.3017 0.2716[4,1,1] 0.3288 0.3398 0.3030 0.3158 0.2924 0.3064[4,2,1] 0.3360 0.3314 0.3107 0.3053 0.3004 0.2941[4,3,1] 0.3340 0.3150 0.3083 0.2869 0.2977 0.2742[4,3,2] 0.3228 0.3290 0.2957 0.3021 0.2842 0.2901[4,3,3] 0.3057 0.3340 0.2773 0.3079 0.2651 0.2965[4,4,1] 0.3266 0.3141 0.3002 0.2861 0.2893 0.2736[4,4,3] 0.3043 0.3344 0.2758 0.3083 0.2635 0.3297Mean 0.3291 0.3280 0.3020 0.3032 0.2909 0.2926


analytical correction terms. It must be emphasized thatthe deviation relative to the benchmark is different forevery interface orientation. For this reason, Figure 4displays the maximum deviation obtained from allinterface orientations, i.e., an upper estimation of thedeviation. The analytical correction term again im-proves the result consistently: without this term, 52 (fcc)and 58 (bcc) shells are necessary to get an accuracy of1 pct. With the analytical correction term, 11 (fcc) and12 (bcc) shells are sufficient. Calculations with 100 shellsplus the correction term will lead to an accuracy of atleast 0.05 pct.

When evaluating the interfacial energy structuralfactor zS,eff/zL,eff as a function of the interface orienta-tion, the orientational dependency is described by thetwo Eulerian angles u and / (Figure 5). Figure 6summarizes the results of the simulations in the formof inverse pole figures (Figures 6(a) and (b)) and u//diagrams (Figures 6(c) and (d)).

Table II summarizes the results of the analysis ofzS,eff/zL,eff for selected low-index interfaces. The minimaof the structural factor zS,eff/zL,eff are found in [110] (bcc)and [111] (fcc) directions; the maxima are found in thedirections of [10,2,1] (bcc) and [10,5,1] (fcc). The resultsare evaluated for different weighting functions, i.e., thepotentials in r�6 to r�8.Table III summarizes the maximum, minimum, and

mean values for an r�6 dependence of the interactionpotential. The mean values were calculated by theintegration of the interfacial energy over the surface of asphere. Apparently, the mean, maximum, and minimumvalues are practically equal for the interfaces in bcc andfcc lattices. Moreover, Table II indicates that thedependence of the interfacial energy from the form ofthe interaction potential is relatively weak, when testingthe potential functions r�6, r�7, and r�8. The positionof the minima and maxima remain the same within therange of the investigated interface orientations. Themean values of the structural factor zS,eff/zL,eff (and,

Fig. 3—Deviation of the lower-index effective coordination numberszL,k from the reference value obtained for 500 nearest-neighbor shellszL,500.

Fig. 4—Maximum deviation of the structural factor zS,eff/zL,eff, rela-tive to the benchmark of 500 nearest-neighbor shells.


thus, the interfacial energy) are decreased by 12 pctfrom 0.3291 (fcc) and 0.3280 (bcc) to 0.2909 (fcc) and0.2926 (bcc), respectively, when modifying the interac-tion potential from r�6 to r�8.The results based on the idealized interaction poten-

tials with r�6, r�7, and r�8 dependence are nowcompared to more realistic potentials adopted fromthe EAM. For example, the second-moment approxi-mations of a tight-binding Hamiltonian are taken. Thesepotentials contain pairwise interactions as well as amultibody term. Both parts can be expressed byexponential functions according to Cleri et al.[47] as

Ec ¼X

i

EiR þ Ei

B

� �½16�

Fig. 5—Orientation of the interface normal vector ~n using u// spacenotation.

Fig. 6—(a) zS,eff/zL,eff for k = 100 in bcc, inverse pole figure; (b) zS,eff/zL,eff for k = 100 in fcc, inverse pole figure; (c) zS,eff/zL,eff for k = 100 inbcc, u// diagram; and (d) zS,eff/zL,eff for k = 100 in fcc, u// diagram.

Table III. Minimum, Maximum, and Mean Values

of zS,eff/zL,eff for 100 Nearest-Neighbor Shells and r26

Dependence of Interaction Potential

Structure Minimum Maximum Mean Anisotropy

Fcc 0.2871 [1,1,1] 0.3398 [10,5,1] 0.3291 18.4 pctBcc 0.2905 [1,1,0] 0.3415 [10,2,1] 0.3280 17.6 pct


where

EiR ¼

X

j

Aab exp �pab rij=rab0 � 1

� �� ½17�

EiB ¼ �

X

j

n2ab exp �2qab rij=rab0 � 1

� �� ( )1=2

½18�

Ec is the total cohesive energy of all atoms i in thesystem containing pairwise bonds (Ei

R) and many-bodyinteractions (Ei

B). Both terms include interactionsbetween atoms i and j. Whereas parameters Aab andnab weight the pairwise and many-body interactions, pand q quantify the decline in the potential function forincreasing bond lengths. Parameters A, n, p, and q arespecific for each atom type. Table IV summarizes theparameters for the potentials for various metals with fccstructure (from Reference 47); Table V presents thecalculated structural factors according to the GBBmodel and using the EAM potentials (Eq. [16]).

All results for the [100], [110], and [111] interfaces arevery close to r�6 potentials, with a mean value of 0.3016(compare r�6: 0.3082), except for Ni (14 pct higher). Thereason for this difference is the slower declining many-body term, with q having a value of only 1.189.

IV. DISCUSSION

This section presents a brief assessment of thereliability of the results of the previous analysis. First,a general evaluation of the theoretical minimum andmaximum interfacial energies is presented. Second, theresults are compared with other broken-bond models.Finally, a comparison is made with published experi-mental and theoretical data.For estimating the minimum and maximum values of

the interfacial energy c, a general potential function r�m

with an a priori undetermined exponent m can be used asa starting point. On investigating the analytical correc-tion term (Appendix II), it is observed that, on the onehand, this term goes to infinity for m £ 4; on the otherhand, however, a minimum is obtained for m fi ¥,which corresponds only to first-nearest-neighbor inter-actions. Consequently, the absolute minima for thestructural factor can be found at zS/zL = 0,2227, for the[110] interface in bcc, and zS/zL = 0,2291, for the [111]interface in fcc lattices. These lower-bound estimatesagree well with the calculated results given in Table III.The absolute minima coincide with regard to theinterface orientation and lie only 7 pct below thefindings for the r�8 potentials.Next, the present GBB model is compared to the two

most advanced broken-bond models: the model of Leeand Aaronson[42] for interfaces and the model ofNicholas[41] for surfaces. Because these authors onlypublish numerical data on the anisotropy and do notprovide absolute numbers for zS/zL, the discussion willbe confined to anisotropies.Lee and Aaronson[42] identify minimum and maxi-

mum energies in fcc for the [111] and [2,1,0] interfaces,respectively. These are in good agreement with theresults presented here. The minimum interface energycoincides exactly with our values, whereas the maximumvalue deviates slightly from [2,1,0] to [2,1,0.1] in thiswork. Also, the isoenergy lines in the inverse pole figure(Figure 6(b)) agree qualitatively. However, the anisot-ropy in the work of Lee and Aaronson is larger: 30 pctcompared in their work to 18 pct in this work. Presum-ably, the reason is the different bond-counting tech-nique: Lee and Aaronson only count the nearestneighbors in specific planes; especially for very distantatom pairs, their counting is incomplete (e.g., bond 3 inFigure 1 would not have been counted). Compared tonear–neighbor atoms, distant atoms are distributedmore isotropically and, thus, lower the anisotropy whenfully considered.Nicholas[41] calculates the anisotropy of surface ener-

gies in fcc and bcc crystal structures. His results aresuitable for comparison when using potential functionsof the Lennard-Jones type f � nr�m �mr�nð Þð Þ withm = 6 and n = 10, 12, or 14, respectively. Again,interface orientations with minimum and maximum cagree well with the results obtained in this work, forboth fcc and bcc. However, Nicholas determined slightlysmaller anisotropies, with 13.2 pct for fcc (18,4 pct inthis work) and 12.1 pct for bcc (17.6 pct in this work).This deviation is partially explained by the secondpotential term r�n; with increasing n, the anisotropy

Table IV. Parameters for EAM Potentials Accordingto Cleri et al.[47]

Element A (eV) n (eV) p q

Ag 0.1028 1.178 10.928 3.139Al 0.1221 1.316 8.612 2.516Au 0.2061 1.790 10.229 4.036Cu 0.0855 1.224 10.960 2.278Ir 0.1156 2.289 16.980 2.691Ni 0.0376 1.070 16.999 1.189Pb 0.0980 0.914 9.576 3.648Pd 0.1746 1.718 10.867 3.742Pt 0.2975 2.695 10.612 4.004Rh 0.0629 1.660 18.450 1.867

Table V. Structural Factors zS,eff/zL,eff for EAM Potentials;

Mean Value Shows Mean Result for [100], [110], and [111]Interfaces

Element [100] [110] [111] Mean

Ag 0.3539 0.2858 0.2378 0.2925Al 0.3633 0.2919 0.2460 0.3004Au 0.3646 0.2823 0.2322 0.2930Cu 0.3582 0.2954 0.2507 0.3014Ir 0.3443 0.2887 0.2419 0.2916Ni 0.4094 0.3447 0.3038 0.3526Pb 0.3612 0.2832 0.2337 0.2927Pd 0.3580 0.2832 0.2338 0.2917Pt 0.3620 0.2824 0.2325 0.2923Rh 0.3597 0.3040 0.2607 0.3081


becomes larger. An additional reason is presumablyfounded in the accuracy of the calculations: Nicholasclaims to have investigated only some thousand bondsper interface orientation. However, an evaluation ofsome of his interfacial orientations requires consider-ation of up to 5 9 107 bonds (this number correspondsto the smallest possible periodic volume element). Mostlikely, the low number of bonds taken into accountintroduced some unknown truncation errors.

Nicholas finally comes to the conclusion that goodresults are achieved already when only including first-and second-nearest neighbors. For some interfaceorientations, this simplification yields correct results;however, the conclusion does not hold generally. Thishas been demonstrated in Figure 4. When including theanalytical correction term, the maximum deviations forc reach up to approximately 10 pct in fcc and bcc (first-and second-nearest neighbors). For a desired deviationbelow 1 pct, at least 11 (fcc) or 12 (bcc) neighbor shells(with the correction term), or 52 (fcc) or 58 (bcc) shells(without the correction term), must be taken intoaccount in the analysis.

Several experimental interfacial energies are availablein the literature from different kinds of precipitatenucleation and coarsening experiments. Saunders andMiodownik[6] compare the interfacial energies obtainedby coarsening experiments of c¢ precipitates in binaryNi-Al systems (experimental data from Calderonet al.,[48] Ardell,[49] Xiao et al.,[50] Wendt et al.,[51] andHirata et al.[52]) and the multicomponent Ni-base alloyNimonic PE16 (experimental data from White[53]). Thescatter of the experimentally obtained values of c iscaused by both the differences in the applied coarseningmodels and the uncertainties in the determination of thediffusion coefficients of the reaction-controlling element.Whereas Monzen et al.[54] obtain interfacial energies forCu particles in bcc iron from coarsening experiments,Stowell,[55] Ardell et al.,[56,57] Yang et al.,[58] Serviet al.,[1] Shiflet et al.,[2] and Robson[5] determine theinterfacial energies of Co precipitates in a Cu matrix bycoarsening or nucleation experiments. Calderon et al.investigated ternary and quaternary alloys.[59] Althoughthere are numerous other experimental data published,only these are taken for comparison, because noCALPHAD-type thermodynamic databases are avail-able for the other systems.

Calculated data on the interfacial energies of c¢precipitates are available from Prince et al.[11] (ab initio

calculations), Staron et al.[20] (cluster dynamics), andMishin[24] and Farkas et al.[60] (EAM combined withfirst principles). Chen et al. calculated the interfacialenergy of the Cr-bcc phase within a Ni-Cr fcc matrix viaEAM.[61] Our model also shows good agreement in thiscase.Table VI compares the data from the literature with

the results obtained with the present GBB model. Goodagreement is achieved in both the binary systems and theternary, quaternary, and multicomponent alloys. Thethermodynamic data for Ni-base alloys are obtainedfrom the commercial database Ni data;[63] The thermo-dynamic data for the other systems are obtained fromthe SSOL4 and TCFE3 databases.[64]

V. CONCLUSIONS

With a newly developed GBB model for multicom-ponent solid solutions, the energy of coherent interfacesand their anisotropy is evaluated in fcc and bcc crystalstructures. The results show good agreement with theexperimental and theoretical values of precipitate/matrix interface energies in both binary and multicom-ponent systems.The mean interfacial energies for both crystal struc-

tures, fcc and bcc, differ only slightly, provided that asufficient number of neighboring shells are taken intoaccount in the analysis of the atomic interactions.Interface orientations with minimum and maximumvalues for c coincide well with other advanced BBmodels in the literature. The anisotropy obtained in thiswork lies between two estimates of alternative models.The GBB model introduced in this work provides a

simple but effective method for obtaining estimates ofcoherent interfacial energies based solely on independentthermodynamic and crystallographic data. It is thusideally suitable for applications in more complex simu-lations of, for example, microstructure/precipitate evo-lution, in which interfacial energies represent an essentialinput quantity and the computational cost is critical.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financialsupport provided by the Osterreichische For-schungsforderungsgesellschaft mbH, the Province of

Table VI. Interfacial Energies from GBB Model in Comparison with Literature Data

Matrix Precipitate T (�C) clit (mJ/m2) cGBB (mJ/m2) Reference(s)

Ni-Al Ni3Al (c¢) 500 to 715 14 to 80 36.2 6, 48 through 52Ni-Al Ni3Al (c¢) �273 to 1300 12 to 63 36.2 11, 20, 24, 60Ni-Cr (fcc) Cr (bcc) 950 200 188 61Fe-Cu Cu (bcc) 600 to 750 520 to 610 570 to 630 54Cu-Co Co (fcc) 450 to 1150 108 to 1008 640 1, 2, 5, 55 through 58Fe-Ni-Al NiAl (B2-bcc) 700 28 to 40 35.7 59Fe-Ni-Al-Mo NiAl (B2-bcc) 700 29 to 42 33.8 59Nimonic PE16 Ni3Al (c¢) 750 90.9 116.4 6, 53Nimonic 80A Ni3Al (c¢) 945 140 99.8 62


Styria, the Steirische WirtschaftsforderungsgesellschaftmbH, and the Municipality of Leoben, within theresearch activities of the Materials Center LeobenForschung GmbH, under the auspices of the AustrianKplus Competence Center Programme.

APPENDIX I

Multicomponent extension of NNBB conceptIn the original approach with atoms A and B, the

derivation of the interfacial energy consists of followingsteps: (1) count the number of broken bonds whendissolving a precipitate phase in an infinite matrix, (2)count the number of broken bonds when introducing aninterface between matrix and precipitate, and (3) com-pare the total energy of the two configurations. Theinterfacial energy in multicomponent systems is derivedanalogously as follows.

Let xMi be the composition of the matrix (i denotesthe element number) and xPj the composition of theprecipitate. When dissolving one mole of precipitatephase in a matrix, all bonds in the precipitate phase(zL/2 bonds per precipitate atom) are first broken. Eachbroken bond in the precipitate is of type i-j with aprobability of xPi x

Pj . In order to dissolve the precipitate

atoms in the matrix, one vacancy is created first in thematrix for each precipitate atom. Again, zL/2 bonds arebroken for each vacancy with each bond of type i-jhaving a probability of xMi xMj . When entering theprecipitate atoms into the matrix, zL new bonds areformed per precipitate atom. When assuming that thematrix is of infinite size, each of these newly formedbonds is of type i-j with a probability of xPi x

Mj . The

total solution enthalpy of one mole of atoms of theprecipitate phase is then (eij denotes the bond energybetween atoms i and j)

DEsol ¼ N�zL�e� ½19�

where

e� ¼X

i

X

j

eij� xMi xPj � xMi xMj =2� xPi xPj =2

� �½20�

The interfacial energy between two phases can becalculated according to the thought experiment intro-duced by Becker.[31] Consider two blocks of matter, oneconsisting of pure matrix phase, the other of pureprecipitate phase. Now break up each block in half andassemble the blocks again after exchanging two halfblocks of matrix and precipitate. By this procedure, oneunit interfacial area of the precipitate and one unitinterfacial area of the matrix phase are broken, whiletwo unit areas of the precipitate/matrix interface arecreated (Figure 7). The interfacial energy is then calcu-lated as the difference in total energy between the twoconfigurations.

If the number of bonds across the interface perinterface atom is denoted by zS and the number ofinterface atoms per square millimeter is nS, zSÆnS/2

bonds are broken in the precipitate and the matrix,respectively. Simultaneously, zSÆnS new bonds are cre-ated for 1 m2 of the precipitate/matrix interface. Theinterfacial energy is then

c ¼ nS�zS�e� ½21�

where

e� ¼X

i

X

j

eij� xMi xPj � xMi xMj =2� xPi xPj =2

� �½22�

Because the result for e* is the same for both cases, thesolution enthalpy and interfacial energy can be setanalogously, in relation to the binary system. Conse-quently, Eqs. [1] and [2] are also valid in generalmulticomponent systems, and Eq. [3] is replaced byEqs. [20] and [22], respectively.

APPENDIX II

Approximation for zS and zL for large distancesCalculation of the number of broken bonds per

interface atom is demonstrated here, assuming a con-tinuums-mechanical solid. The starting point is thenumber of broken bonds per interface area A (Figure 8):

dbeff ¼ nV�dVa�nV�dVb�fðr=r1Þ ½23�

This number is normalized to broken bonds per inter-face atom, according to Eq. [11]. Substituting nS withnV leads to

dzS;corr ¼ n4=3V �2r2p� 1� cos/ð Þ�fðr=r1Þ�da�db ½24�

The terms / and r can be expressed in terms of a andb with

dzS;corr ¼ n4=3V �2p� aþ bð Þb�f aþ bð Þ=r1ð Þ�da�db ½25�

with f as a potential function

f ¼ aþ bð Þ=r1½ ��m ½26�

Because the correction term involves pairwise atomicbonds only in distances larger than rk, the constrainta + b ‡ rkholds. Accordingly, zS,corr can be writtenas

Fig. 7—A block of matrix and a block of precipitate is exchanged tocreate the precipitate/matrix interface.


zS;corr ¼ 2p�n4=3V �rm1 �Z1

aþb�rk

aþ bð Þ�mþ1b�da�db ½27�

which has the solution

zS;corr ¼ 2p�n4=3V �rm1 �r4�mk

m� 2½28�

For zL,corr, the number of bonds with distance r is

dzL;corr ¼ nV�4pr2�dr�f r=r1ð Þ ½29�

Assuming the potential function given in Eq. [26] andonly considering bonds with a length r ‡ rk delivers

zL;corr ¼ nV�4p�rm1Z1

r¼rk

r�mþ2dr ½30�

which has the solution

zL;corr ¼ nV�4p�rm1 �1

m� 3r3�mk ½31�

REFERENCES1. I.S. Servi and D. Turnbull: Acta Metall., 1966, vol. 14, pp. 161–69.2. G.J. Shiflet, Y.W. Lee, H.I. Aaronson, and K.C. Russell: Scripta

Metall., 1981, vol. 15, pp. 719–22.3. R. Kampmann and R. Wagner: Acta Metall., 1984, pp. 91–103.4. J.D. Robson: Acta Mater., 2004, vol. 52, pp. 4669–76.5. J.D. Robson: Mater. Sci. Technol., 2004, vol. 20, pp. 441–48.6. X. Li, N. Saunders, and A.P. Miodownik: Metall. Mater. Trans.

A, 2002, vol. 33A, pp. 3367–73.7. A.J. Ardell, R.B. Nicholson, and J.D. Eshelby: Acta Metall., 1966,

vol. 14, pp. 1295–1309.

8. A.D. Brailsford and P. Wynblatt: Acta Metall., 1979, vol. 27,pp. 489–97.

9. P.W. Voorhees: J. Stat. Phys., 1985, vol. 38, pp. 231–52.10. K. Tsumuraya and Y. Miyata: Acta Metall., 1983, vol. 31,

pp. 437–52.11. D.L. Price and B.R. Cooper: Mater. Res. Soc. Symp. Proc., 1996,

vol. 408, pp. 463–68.12. K. Rapcewicz, B. Chen, B. Yakobson, and J. Bernholc: Phys. Rev.

B, 1998, vol. 57, pp. 7281–91.13. M. Sluiter and Y. Kawazoe: Phys. Rev. B, 1996, vol. 54,

pp. 10381–10384.14. M. Sluiter, M. Asta, and Y. Kawazoe: Sci. Rep. Res. Inst. Toh.

Univ. Ser. A, 1006, vol. 41, pp. 97–101.15. J. Hartford: Phys. Rev. B, 2000, vol. 61, pp. 2221–29.16. R. Benedek, D.N. Seidman, and C. Woodward: Interface Sci.,

2004, vol. 12, pp. 57–71.17. A. Landa, P. Wynblatt, E. Johnson, and U. Dahmen: Acta Mater.,

2000, vol. 48, pp. 2557–63.18. P. Wynblatt: Appl. Surf. Sci., 2003, vol. 219, pp. 39–46.19. F. Ercolessi, M. Parrinello, and E. Tosatti: Philos. Mag. A, 1988,

vol. 58, pp. 213–26.20. P. Staron and R. Kampmann: Acta Mater., 2000, vol. 48, pp. 701–

12.21. M. Asta: Acta Mater., 1996, vol. 44, pp. 4131–36.22. M. Asta, V. Ozolins, and C. Woodward: JOM, 2001, vol. 53,

pp. 16–19.23. M. Asta and J.J. Hoyt: Acta Mater., 2000, vol. 48, pp. 1089–96.24. Y. Mishin: Acta Mater., 2004, vol. 52, pp. 1451–67.25. J. Svoboda, F.D. Fischer, P. Fratzl, and E. Kozeschnik: Mater.

Sci. Eng., A, 2004, vol. 385, pp. 166–74.26. E. Kozeschnik, J. Svoboda, P. Fratzl, and F.D. Fischer: Mater.

Sci. Eng., A, 2004, vol. 385, pp. 157–65.27. E. Kozeschnik, J. Svoboda, and F.D. Fischer: CALPHAD, 2005,

vol. 28, pp. 379–82.28. J.D. Robson, M.J. Jones, and P.B. Prangnell: Acta Mater., 2003,

vol. 51, pp. 1453–68.29. N. Saunders and A.P. Miodownik: Calphad, CALPHAD—

A Comprehensive Guide, Pergamon Materials Series, R.W. Cahn,ed., Elsevier, Oxford, United Kingdom, 1998, vol. 1.

30. W.L. Bragg and E.J. Williams: Proc. R. Soc., 1934, vol. 145,pp. 699–730.

31. R. Becker: Ann. Phys., 1938, vol. 32, pp. 128–40.32. D. Turnbull: Impur. Imperf., 1955, pp. 121–43.33. E.A. Guggenheim: Trans. Faraday Soc., 1945, vol. 41, pp. 150–56.34. I. Prigogine and R. Defay: Bull. Soc. Chim. Belg., 1950, vol. 59,

pp. 255–62.35. S. Ono: Mem. Fac. Eng., 1947, vol. 10, pp. 195–254.36. M. Hillert: Doctoral Thesis, Massachusetts Institute of Technol-

ogy, Cambridge, MA, 1956.37. M. Hillert: Acta Metall., 1961, vol. 9, pp. 525–35.38. J.W. Cahn and J.E. Hilliard: J. Chem. Phys., 1958, vol. 28,

pp. 258–67.39. M. Kahlweit: Z. Phys. Chem. N.F., 1962, vol. 34, pp. 163–81.40. J.L. Meijering: Acta Metall., 1966, vol. 14, p. 251.41. J.F. Nicholas: Aust. J. Phys., 1968, vol. 21, pp. 21–34.42. Y.W. Lee and H.I. Aaronson: Acta Metall., 1980, vol. 28, pp. 539–

48.43. Z.G. Yang and M. Enomoto: Metall. Mater. Trans. A, 2001,

vol. 32A, pp. 267–73.44. Z.G. Yang and M. Enomoto: Mater. Sci. Eng., A, 2002, vol. 332,

pp. 184–92.45. A.R. Miedema and F.J.A. den Broeder: Z. Metallkd., 1979,

vol. 70, pp. 14–20.46. J.E. Lennard-Jones: Proc. Phys. Soc., 1931, vol. 43, pp. 461–82.47. F. Cleri and V. Rosato: Phys. Rev. B, 1993, vol. 48, pp. 22–33.48. H.A. Calderon, P.W. Vorhees, J.L. Murray, and G. Kostorz: Acta

Metall., 1994, vol. 42, pp. 991–1000.49. A.J. Ardell: Acta Metall., 1968, vol. 16, pp. 511–16.50. S.Q. Xiao and P. Haasen: Acta Metall., 1991, vol. 39, pp. 651–59.51. H. Wendt and P. Haasen: Acta Metall., 1983, vol. 31, pp. 1649–59.52. T. Hirata and D.H. Kirkwood: Acta Metall., 1977, vol. 25,

pp. 1425–34.53. R.J. White: Mater. Sci. Eng., 1978, vol. 33, pp. 149–57.54. R. Monzen, K. Takada, and K. Matsuda: Z. Metallkd., 2003,

vol. 94, pp. 1241–46.

Fig. 8—Volume elements in phase a and b causing broken bonds ina distance r.


55. M.J. Stowell: Mater. Sci. Technol., 2002, vol. 18, pp. 139–44.56. R.A. Oriani: Acta Metall., 1964, vol. 12, p. 1399.57. A.J. Ardell: Acta Metall., 1972, vol. 20, p. 61.58. S.C. Yang and P. Nash: Mater. Sci. Technol., 1988, vol. 4,

pp. 860–66.59. H. Calderon and M.E. Fine: Mater. Sci. Eng., 1984, vol. 63,

pp. 197–208.60. D. Farkas, M.F. de Campos, R.M. de Souza, and H. Goldenstein:

Scripta Metall. Mater., 1994, vol. 30, pp. 367–71.

61. J.K. Chen, D. Farkas, and W.T. Reynolds, Jr.: Acta Mater., 1997,vol. 45, pp. 4415–21.

62. R.A. Ricks, A.J. Porter, and R.C. Ecob: Acta Metall., 1983,vol. 31, pp. 43–53.

63. Thermodynamic Database Ni-Data, version 5, Thermotech, Ltd.,Surrey, United Kingdom.

64. Thermodynamic Databases SSOL4, version 4, and TCFE3, version3, ThermoCalc AB, Stockholm.


Generalized Nearest-Neighbor Broken-Bond Analysis of Randomly Oriented Coherent Interfaces in...

Documents

Transcript of Generalized Nearest-Neighbor Broken-Bond Analysis of Randomly Oriented Coherent Interfaces in...