Nonempirical simulations of boron interstitials in tungsten

�On sabbatical leave from Ben-Gurion University of theNegev, Beer-Sheva, Israel.

*Corresponding author. Department of Materials Engineer-ing, Ben-Gurion University of the Negev, P.O. Box 653, 84105Beer-Sheva, Israel. Tel.: #972-7-461460; fax: #972-7-472946.E-mail address: [email protected] (D. Fuks).

Physica B 301 (2001) 239}254

Nonempirical simulations of boron interstitials in tungsten

Kleber C. Mundim�, Vlad Liubich�, Simon Dorfman�, Joshua Felsteiner�,David Fuks*��, Gunnar Borstel�

�Instituto de Quimica, Universidade de Brasilia, Caixa Postal 4478, 70919-970 Brasilia, Brazil�Department of Materials Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel

�Faculty of Physics, Israel Institute of Technology } Technion, 32000 Haifa, Israel�Department of Physics, University of Osnabrueck, D-49069 Osnabrueck, Germany

Received 15 September 2000; received in revised form 1 December 2000

Abstract

Formation of W}B solid solutions for di!erent concentrations of boron is studied within nonempirical modeling. Weconsider ordering tendencies, study electronic structure and provide total energy calculations on the basis of coherentpotential approximation. We also study an equilibrium structure of a lattice with �

��1 1 1� grain boundary in pure

tungsten and in tungsten-based solid solution with boron additives. We used simulated annealing methods in atomisticsimulations to obtain relaxed con"gurations of the lattice in the vicinity of grain boundary. � 2001 Elsevier ScienceB.V. All rights reserved.

Keywords: Boron interstitials; Tungsten; Nonempirical simulations

1. Introduction

Large cohesive energy and bulk modulus oftungsten cause wide use of this refractory metal inthe industry and are the topic of a lot of investiga-tions in material science. Measurement of proper-ties of W is a serious problem because of theextremely low compressibility. That is why anypredictions of the behavior of tungsten have to be

appreciated. The nonempirical calculations oftungsten were used to study the in#uence of micro-alloying on the ductile}brittle phase transforma-tion [1]. Impurities such as N, O, P, S, and Si,weaken the intergranular cohesion in Fe and W re-sulting in &loosening' of the grain boundary (GB).The presence of B and C, on the contrary, enhancesthe interatomic interaction across the GB. Boronplays a dual role in both Fe and W; not only doesits presence at GBs enhance the intergranular cohe-sion, but it also accomplishes `site competitioncleansinga (SCC) by displacing the other impurityatoms o! the GB. Microalloying with 10}50ppmB may be an e!ective way of improving the ductil-ity of both Fe-based alloys and W [1]. The borone!ects in metals and alloys are typically attributedto its intergranular segregation. This hypothesis

0921-4526/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 2 7 9 - 4

comes from experimental measurements of inter-granular boron enrichment, mainly in Ni

�Al alloys,

by Auger electron spectroscopy (AES) [2]. Theformation and stability of these segregations is dif-fusion controlled and demands a knowledge of the"ne structure of the interface between two parts of`perfecta crystal which are in contact at the grainboundary. Interatomic interactions determine anactivation energy of the interstitial atom di!usionprocess. The activation energy strongly depends onthe local con"guration of the lattice in the vicinityof interstices. For example, in Ref. [3] it was shownthat the height of potential barrier, �;, for inter-stitial carbon di!usion in copper for the rigid latticeis 0.99 eV and for the relaxed lattice is 1.09 eV. Thissmall change in �; leads to drastic changes in therate of di!usion process. The ratio of di!usioncoe$cients in the relaxed and rigid lattices at tem-peratures about 800K is 3.17. Thus displacementslead to a substantial decrease of di!usivity in theseinterstitial solid solutions. Analogous e!ects maybe expected for GB di!usion. The reconstruction ofthe lattice near the GB is important also for under-standing the fundamental physics of the processof formation of the intergranular segregations.A number of papers were devoted to the modelingof tungsten GBs in di!erent approaches (see, forexample, Refs. [4,5] and references therein). How-ever, to the best of our knowledge no systematicstudy of the in#uence of additives on the recon-struction of the lattice near GB in W is available inthe existing literature. Even less is known about thereconstruction of the structure when an impurityoccupies the interstitial position at GB.In the framework of the density functional theory

(DFT) conditions of formation of W}B solid solu-tions for di!erent concentrations of boron arestudied. On the basis of coherent potential approxi-mation (CPA) we consider ordering tendencies,study electronic structure and provide total energycalculations. We also study an equilibrium struc-ture of a lattice with �

��1 1 1�GBwhen the system

is relaxed in pure tungsten and the same with boronadditives, which occupy the interstitial position atthe GB. This gives microscopic information on howthe elastic "eld is spread near the GB. Such in-formation forms the basis for the detailed model-ling of a di!usion process that leads to the

formation of the segregations at the GBs. Our the-ory in Ref. [3] was based on concentration waveapproach and on the linear approximation in lat-tice dynamics. For GB this approach is completelyunapplicable. This is a reason to use simulatedannealing methods [6,7] in atomistic simulationsto obtain relaxed con"gurations of the lattice in thevicinity of GB.

2. Stochastic dynamics: generalized simulatedannealing

Simulated annealing [6,7] methods have beenapplied successfully in the description of a varietyof global extremization problems. These methodshave attracted signi"cant attention due to theirsuitability for large-scale optimization problems,especially for those in which a desired global min-imum is hidden among many local minima. Thebasic aspect of the simulated annealing method isthat it is analogous to thermodynamics, especiallyconcerning the way in which liquids freeze andcrystallize, or in which metals cool and anneal. Athigh temperatures, the atoms move freely with re-spect to one another. If the system is cooled slowly,thermal mobility is lost. The atoms, say in complic-ated molecules, are often able to line themselves upand assume a molecular geometry that is in generala local equilibrium state. The simulated annealingprocedure is actually more complicated than thecombinatory one, since the familiar problem oflong, narrow potential valleys again asserts itself.Simulated annealing, as we will see, tries randomsteps, but in a long, narrow valley, almost all ran-dom steps are uphill. The amazing fact is that, fora slowly cooled system, nature is able to "nd thisminimum energy state. So the essence of the processis slow cooling, allowing sample time for redistribu-tion of the atoms as they lose their mobility. This isthe technical de"nition of annealing, and it is essen-tial for ensuring that the lowest energy state will beachieved.The "rst nontrivial solution along this line was

provided by Kirkpatrick et al. [6,7] for classicalsystems, and also extended by Ceperley et al. [8] toquantum systems. It strictly follows the quasi-equilibrium Boltzmann}Gibbs statistics using a

240 K.C. Mundim et al. / Physica B 301 (2001) 239}254

Gaussian visiting distribution, and is sometimes re-ferred to as classical simulated annealing (CSA) orthe Boltzmann machine. The next interesting step inthis subject was Szu's proposal [9] to use a Cau-chy}Lorentz visiting distribution, instead of a Gaus-sian distribution. This algorithm is referred to as thefast simulated annealing (FSA) or Cauchy machine.On the other hand, a generalized simulated an-

nealing (GSA) approach which closely follows therecent Tsallis statistics [10}12] has been proposed[13}16]; GSA includes both the FSA and CSAprocedures as special cases. We have implementedthe GSA algorithm as a method to calculate theminimum energies of conformational geometriesfor di!erent molecular structures. This techniquecan be applied in either quantum [14] or classical[15] methods. The GSA method is based on thecorrelation between the minimization of a costfunction (conformational energy) and the geomet-ries randomly obtained through a slow cooling. Inthis technique, an arti"cial temperature is introduc-ed and the system is gradually cooled in completeanalogy with the well known annealing technique,frequently used in metallurgy when a molten metalreaches its crystalline state (global minimum of thethermodynamic energy). In our case the temper-ature is intended as an external noise, which acts asa convenient stochastic source for eventual detrap-ping from local minima. Towards the end of theprocess the system hopefully is within the attractivebasin of the global minimum. The challenge is tocool the system as fast as possible and still have theguarantee that no irreversible trapping at any localminimum has occurred. More precisely, we searchfor the quickest annealing (approaching a quench-ing) which maintains the probability of "nishingwithin the global minimum close to one.The procedure to search the minima (global and

local) or to map the energy hypersurface consists incomparing the conformational energy for two con-secutive random geometries x

��and x

�obtained

from the GSA routine. x�is an N-dimensional vec-

tor that contains all atomic coordinates (N) to beoptimized.The geometries, for two consecutive steps, are

related by

x��

"x�#�x

�, (1)

where �x�is a random perturbation on the atomic

position.To generate the random vector �x

�the present

GSA routine uses an extension of the proceduregiven in Ref. [15]. We have calculated�x"g��(�)using a numerical integration of the visiting distri-bution probability g

��(x), where � is a random

vector [0, 1] obtained from an equiprobability dis-tribution and g�� is the inverse of the integral ofg��(x) given by

g��(�)"inverse��

��

g��(x) dx�. (2)

Mathematical details of the structure of the distri-bution function g and its inverse g�� are given inRef. [15]. In summary, the complete algorithm formapping and searching for the global minimum ofthe energy is as follows:

(i) Fix the parameters (q�;q

�). We note that

(q�;q

�)"(1;1) and (1;2) respectively correspond

to the Boltzmann and Cauchy machines. Startat t"1, with arbitrary internal coordinatesand high enough value for visiting temperature¹

��(1) and cool as follows:

¹��(t)"¹

��(1)

2��!1

(1!t)��!1, (3)

where t is the discrete time corresponding tosteps of computer iteration.

(ii) Next, randomly generate the new atomic coor-dinate x

��from x

�as given by the visiting

distribution probability g��as follows:

x��

"x�#g��(�). (4)

For su$ciently long time simulations this pro-cedure assures that the system can both escapefrom any local minimum and explore the entireenergy hypersurface. This equation is used inthe GSA routine and di!ers from the generalproposal given in Refs. [11,12]; instead webuild a minimization vector using Eq. (1).

(iii) Then calculate the conformational energyE(x

��) from the new molecular geometry

using the classical force "eld [16]. The new

K.C. Mundim et al. / Physica B 301 (2001) 239}254 241

P��

"(x�Px

��)"�

1 ifE(x��))E(x

�),

1

1#[1#(q�

!1)(E(x��)!E(x

�))/¹

��(t)]��

ifE(x��)'E(x

�).

(5)

Fig. 1. Schematic diagram of generalized simulated annealingprocess.

energy value will be accepted according to thefollowing rules:

ifE(x��))E(x

�), replacex

��by x

�;

ifE(x��)'E(x

�), run a randomnumber

r3[0, 1];

if r'P��(acceptance probability) retain x

�;

otherwise, replacex�byx

��.

The acceptance probability is given by

(iv) Calculate the new temperature ¹��(t) using Eq.

(5) and go back to (ii) until the convergence ofE(x

�) is reached within the desired precision.

In order to clarify the procedure to construct thepresently used computational code, we present the#owchart in Fig. 1.

3. Selection of interatomic potentials

Large-scale atomistic simulations are necessaryin the study of complex physical phenomena suchas fracture, plastic deformation, friction, etc. It is ofcourse the challenge of the theory to present theinteratomic interactions in these simulations ina framework of quantum mechanical approach. Inthis case one has to consider the behavior of theelectronic system at each step of calculations. How-ever, till now such an approach could be appliedonly to systems that include a relatively small num-ber of atoms (of order 10). The alternative way isthe treatment of the system in terms of e!ectiveinteratomic potentials that allows fast determina-tion of the energies, thus making it possible to studythe systems that involve much more atoms.In most cases, including those where the struc-

tural energy di!erences are important, the

long-range interaction potentials are necessary todescribe accurately a wide range of local atomicenvironments. This concerns also the study of thegrain boundary relaxations, which we are going tocalculate.To obtain such potentials an exact procedure for

inverse ab initio energy data will be used here. Theinversion approach was "rst formulated in Ref.[17] for the pair potentials and for many-bodypotentials in Ref. [18].


3.1. LDA model

The essential requisite for such a research is tohave a reliable and e$cient electronic structuremethod to calculate the total energies and elec-tronic structures of the interstitial solid solutions intungsten. Within the framework of the Hohen-berg}Kohn}Sham density functional theory theelectronic structure can be e$ciently handled usinga "rst-principles self-consistent method. DFT haswide applications to molecules and solids [19].Even the crudest approximation, local density ap-proximation (LDA), to the density functional the-ory has been successfully applied to predictstructural and dynamic properties of a large varietyof materials. Equilibrium volumes, elastic con-stants, phonon frequencies, surface reconstruction,magnetism are just some examples of propertieswhich could be successfully calculated for systemswithout particularly strong electron correlationswithin the LDA. The LDA usually leads to someoverbonding in solids (equilibrium volumes aretypically 1}3% underestimated). Considerably lar-ger errors are found in cases where the LDA is notsu$ciently accurate; ionic compounds like MgOserve as examples when the simple LDA fails.The electronic structure of crystalline solids

could be e$ciently calculated using the linear bandstructure methods. The linear mu$n-tin orbitals(LMTO) method is particularly fast and e$cientfor handling complex and large unit cells because ofthe ease with which the structure-dependent partand the potential-dependent part are separated outin the secular equation [20,21]. An important mile-stone in the application of the LMTO was tounderstand that the original in"nite LMTO basisset can be limited only by few orbital functions.With this advantage the LMTO method has thecomputational simplicity of the empirical tight-binding schemes, as well as the accuracy of other"rst principles methods.LMTO in the atomic sphere approximation

(ASA) approach is a well-established technique tostudy metals, alloys, perovskites and di!erentphases with nontrivial structures [22}25]. Forclose-packed metallic structures, in particular, thismethod consistently predicts reliable and accurateresults comparable to those obtained from other

sophisticated LDA-based methods like linearizedaugmented plane wave (LAPW) [26]. In the lastfew years LMTO ASA method was successfullyapplied also to the study of surface alloys (see, forexample Refs. [27,28]). In the case of dilute alloysthe properties of Fe embedded in V and Crmatriceswere studied in Ref. [25]. It seems that today thisscheme is one of the most promising techniques inband structure studies due to the ability to calcu-late fast very complicated structures. The LMTOASA method is ideally suited to the relativelyclose-packed solid solutions of tungsten treatedhere, where one can ensure a reasonably smalloverlap between the atomic spheres. We managedthe value of this overlap by introducing an addi-tional interstitial (&empty') sphere. In LMTO ASA,the approximation due to spherical averaging isalso manageable by changing the ratio betweenatomic spheres of di!erent species. The overlapbetween the spheres is de"ned as (S

�#S

!d)�

100/2S�and is less than 30%. Here S

�and S

(S

�(S

) are the radii of the two overlapping

spheres and d is the distance between them. Incor-porating the so-called &combined corrections', onecan partly salvage the error due to spheridization ofpotential and charge density [20,21].Band structure calculations based on the den-

sity-functional theory allow to obtain a quantitat-ive description of the ground-state properties ofabsolutely ordered alloys. Application of thesemethods to the calculations of the thermodynamicproperties of partially ordered or random alloysgives reasonable results [27}33]. The most attract-ive feature of the single-site (SS) coherent potentialapproximation (CPA) is the ability to apply thisscheme to the direct calculations of the electronicstructure of randomly or partially ordered alloys[29,30]. Recent applications of the CPA schemeshow that this method allows to accurately repro-duce lattice parameters, bulk moduli, and enthal-pies of formation (see, for example, Refs. [27,28]and references therein). This accuracy is similar tothe accuracy of other local density-functionalmethods for completely ordered phases.In order to study the e!ects of the electronic

density distributions in tungsten-based solid solu-tions on chemical bonding we model interstitialsolid solutions with a cell shown in Fig. 2. Behavior


Fig. 2. Structure of a cell, which was used in CPA-LMTOsimulations. Numbers 1 and 2 de"ne the sub-sublattices ofoctahedral interstitial positions of the "rst sublattice, sub-sub-lattices of the second sublattice are numbered as 3 and 4, andthose of the third sublattice are numbered as 5 and 6. Thecrossed circles show tungsten atoms.

Fig. 3. Total density of states (in number of states per atom per Ry) in interstitial W}B solid solutions for di!erent atomic fractions ofboron. E

�in the picture is the Fermi energy, c

�is the atomic fraction of boron. A solid solution is studied for the ordered state with the

partial occupation of only one sublattice.

of boron atoms in tungsten is studied on the basisof the "rst-principles total energy calculations ofbinary, partially ordered compounds in the frame-work of the CPA. The calculations have been per-formed in the scalar-relativistic approach fora number of di!erent volumes per atom. Core elec-trons were frozen after initial atomic calculations.All the calculations were done in the framework ofthe ASA [20,21]. The individual atomic sphereradii of W, B, and empty (E) spheres were set in theratio 1 : 0.6 : 0.6, respectively. The convergence cri-terion for the total energy was 0.001mRy. Typi-cally, 60 iterations were needed to achieve thenecessary convergence. The equilibrium latticeparameter and corresponding ground-state energyof a given alloy were obtained on the basis of a setof self-consistent calculations of the total energyclose to the equilibrium lattice parameter with a "tto a Morse-type equation of state [34]. In Fig. 3 weshow plots of the total density of states (DOS) forW}B interstitial solid solutions with di!erent con-centrations of B placed in position 1 of the "rstsublattice (see Fig. 2). DOSs correspond to di!erent


compositions of interstitial W}B solid solutionswith the equilibrium lattice parameter. The integra-tion over the Brillouin zone has been performedwith the special points technique (the number ofpoints was about 250).We present herewith only a brief account of the

technique of the usual SS CPA for solids withseveral di!erent sublattices [31]. Let us consider aninterstitial BCC solid solution W

�B��, where

B occupies the octahedral interstitial sites of BCClattice and c is the atomic fraction of W. There arethree equivalent BCC sublattices of such interstitialsites. If on each of these sublattices, numerated byp(p"1, 2, 3), N

�sites of all N sites of a sublattice

are occupied by boron atoms then c�"N

�/N is the

atomic fraction of boron on the corresponding sub-lattice and

c"

N

N#

��

N�

"

1

1#��

��

c�

(6)

To study the ordering in the interstitial solid solu-tion it is convenient to determine the probabilityn�(r) to "nd an interstitial atom in the position r on

the pth sublattice. This ordering is considered asthe ordering between interstitial atoms and`emptya sites in each sublattice. It is worth men-tioning that actually two types of ordering mayoccur in the case when there exist several equivalentsublattices of interstitial sites. One type of orderingcorresponds to the situation when, for example,interstitial atoms occupy only one (say, p"1) sub-lattice of interstitial positions, leaving all the othersublattices empty (n

"n

�"0). In this case the

interstitial atoms may be still randomly distributedon the sites of this sublattice. Decrease of the tem-perature may result in the ordering on this sublat-tice, and in this situation n

��(r) takes several

constant values. Each value de"nes sub-sublatticeof the ordered phase, which is formed on the "rstsublattice of interstitial sites. If, in the simplest case,a B2-type ordered phase is formed on the "rstsublattice of interstitial sites then for a stoichiomet-ric composition (c

�"�

)

n��

"c�#�

�"1 (for the corners of the bcc cubic

cell of the "rst interstitial sublattice),

n��

"c�!�

�"0 (for the centersof the same cell),

here the upper index numerates sublattices, and � isthe long-range order parameter, which is de"nedon the "rst sublattice in the usual manner:�"(n��

�!c

�)/(1!c

�) (see, for example, Ref. [35]).

The probabilities to "nd empty sites are de"ned as(1!n

�), and the corresponding equations for them

may also be written.Now we are ready to rede"ne the probabilities

n�(r) in the vector form. Components of this vector

determine the occupation of the octahedral sublat-tices by interstitial atoms. The vector, which corres-ponds to the formation of B2-type phase on the "rstsublattice with the second and the third sublatticesbeing empty is given by n"�n��

��2� as

n"(c�#�

�, c

�!�

�, 0, 0, 0, 0). (7)

For example, for this speci"c case (c"c

�"0) for

the atomic fraction of boron equal to 0.01 and theatomic fraction of tungsten equal to 0.99 the corre-sponding value c

�may be found from the equation

0.99"1

1#c�

,

that gives c�"0.010101 and for the atomic frac-

tion of B equal to 0.167 the value c�"0.20048.

The average one-electron Green's functionshould be determined for calculations of electronicstructure and ground state properties. To obtainthis function we apply the SS CPA in conjunctionwith the LMTO method in the atomic sphere ap-proximation (ASA). The average one-electronGreen's function may be obtained in the form of theKorringa}Kohn}Rostoker (KKR) ASA Green'sfunction, which is identical to the scattering pathoperator in multiple scattering theory. For a com-plex energy z we have

��(z)"

1

<��

d�k[R(z)!�(k)]��, (8)

where <��is the volume of Brillouin zone (BZ),

�(k) is the LMTO structure constant and R(z) isthe crystal coherent-potential matrix. The sub-scripts i and j refer to individual sub-sublattice sitesin the unit cell. We have omitted the angular mo-mentum quantum numbers (l

�) as well as LMTO

representation number. The coherent-potential


matrix that enters Eq. (8) is block-diagonal:

R"�R

�0 2 0

0 R 2 0

2 2 2 0

0 0 2 R� (9)

and each diagonal element R�is the coherent-po-

tential function of the ith sub-sublattice. To obtainthe complete coherent-potential function we mustsolve for the corresponding equations of each sub-sublattice, which for our case are

R�"n��R�

�#(1!n��)R�

�

#[R��!R�

�]�

��[R�

�!R�

�]�� , (10)

R�"R�

�#R�

��R�

�� ,

where n�� is the occupation probability for boronatoms on the ith sub-sublattice, andR�

�andR�

�are

the coherent-potential functions of B and E species,respectively. Here E stands for an empty interstitialsite, or `emptya sphere in ASA. Coherent-potentialfunctions are coupled by the de"nition (Eq. (8)) ofthe coherent Green's function, which, together withEq. (10), forms the nonlinear system of CPA equa-tions that must be solved self-consistently.We calculated electronic density distributions

with the CPA LMTO code [27,28,31]. This codeincludes the determination of the Madelung-energyprefactor , which makes the CPA LMTO resultsagree with those obtained by the Connoly}Will-iams method on the basis of the total energies ofordered alloys [29]. This prefactor enters into theexpression for the Madelung energy of the alloyand for the ith sub-sublattice of tungsten-basedinterstitial solid solution it is

E��

"!en��(1!n��

�)(Q

�!Q

�)

R��(1). (11)

Here R��(1) is the radius of the "rst coordinationshell of the ith sub-sublattice, e is the electroncharge, boron and empty sphere charges are Q

�and Q

�, respectively. These charges are de"ned as

Q�"�

��

�d�r!Z

�and Q

�"�

��

�d�r.

In these equations S��

is the radius of an atomicsphere, Z

�is the atomic number of boron, and

�and

�are the electronic densities of a boron

atom and an empty sphere, respectively. A numberof models could be reduced to this equation [32].

3.2. Interatomic potentials from nonempiricalcalculations

The scheme of calculations of nonempirical in-teratomic potentials for pure metals, which is usedin our paper, includes the following steps:

(1) LMTO CPA calculation of the total energiesof pure tungsten and boron in BCC lattice fordi!erent lattice parameters, a. The obtained valuesare "tted to the Morse-type function

E��

"A#Be�� !2Be��.

It is easy to see that the last two terms present thecohesive energies of the pure components. The de-mand that both components have to be calculatedin the same (BCC) lattice is dictated by the de"ni-tion of the mixing energy where all the energieshave to be given in the same crystalline structure asthe alloy under investigation.(2) With the obtained dependence of cohesive

energy on the distance in the simplest case of pairpotentials we present this energy by

E(r)"��

��

n�<(s

�r) (12)

with atomic separation grouped into coordinationshells p of radius s

�r, containing n

�atoms each.

Uniform dilatation of the lattice expresses itself invarying the parameter r with the structural quantit-ies �s

�� and �n

�� "xed. Shells are numbered so that

s�(s

(s

�(2, and distances are scaled so that

s�"1. The desired inversion formula for V[E] may

be obtained from Eq. (12) by rearrangement of theterms

<(r)"1

n��E(r)!

��

��

n�<(s

�r)�. (13)


Recursive substitution now generates the explicitformula

<(r)"1

n�

E(r)!��

n�

n�

E(s�r)

#

��

n�n�

n��

E(s�s�r)!2 . (14)

This way we obtain the ab initio e!ective pairpotentials for pure tungsten and boron. The prob-lem of calculation of interatomic potentials, whichare able to describe an interaction between di!erentspecies is more complicated than extractingthe interatomic potentials for pure elements.The investigation is implemented in the followingstages:1. For extremely dilute W}B solid solutions we

perform the total energy calculations for the binarydisordered alloy and its components with the linearmu$n-tin orbitals (LMTO) method in the frame-work of the coherent potential approximation(CPA) (CPA LMTO code [27,28,31]). In this ap-proach the total energies are calculated for varyinglattice constants in order to determine the equilib-rium lattice constant and to obtain the dependenceof the energy on the interatomic distance.2. These values of the energies are used to de"ne

the mixing energy, �E, as a function of the latticeparameter

�E"E��

![cE�

#(1!c)E�], (15)

where c is the atomic fraction of the component A,E�and E

�are the energies of the components. In

the approximation of pairwise interactions theseenergies are

E�

"

1

2��

<��(�r

�!r

��),

E�"

1

2��

<��(�r

�!r

��). (16)

3. Making use of the regular solid solution model

�E";c(1!c) (17)

we can now present the sum of A}B interatomicpotentials as in Ref. [35]:

�E"��

<��(�r

�!r

��)";#E

�#E

�.

Although this result is a simple one it is very impor-tant because it allows to extract A}B potentialsfrom ab initio calculations. To obtain such poten-tials an exact procedure for inverting ab initio en-ergy data is used [17,18,36].4. Using Eq. (15) we calculate the dependence of

the parameter; from Eq. (3) on the distance, whichaccording to Eq. (17) gives also the dependence of�E on the lattice parameter. Keeping in mind Eq.(17) the e!ective interatomic potentials <

��(r) are

calculated with the recursion formula (14), wherewe substitute �E(r)/2 for E(r).

4. CPA calculations

Our study is based on the analysis of electronicdensity distributions for di!erent interatomic dis-tances, supercell con"gurations, and compositionsof an interstitial impurity. Changes in the concen-tration will lead to changes of the supercell volumeand to changes in the character of the bondingforces. Band structures of completely ordered tung-sten-based interstitial alloy with partially occupiedoctahedral positions 1 and 2,

site 1N(��0) and site 2N(0 0 �

),

were carried out within the CPA LMTO procedurebrie#y outlined in the previous section (for detailssee Refs. [27,28,31]). With the self-consistently ob-tained bands we calculated the equilibrium totalenergies of the completely ordered phase andphases with partially occupied sublattices.In our calculations we studied the behavior of

the dilute solid solutions of boron in tungsten.According to the Hagg rule we assumed that thesesolutions are interstitial, because the ratio ofatomic radii of constituents is less than 0.49. Boronatoms were distributed on one of three sublatticesof octahedral interstitial positions of the hostmatrix. Such a modeling had two main aims: (a) tostudy the tendencies of ordering of boron atoms on


Table 1Total number of states, ¹

��(in number of states per atom per Ry) for tungsten and boron atoms at the Fermi energy, E

�, and total

energy, E��, per cell for W}B dilute solid solutions

Atomic fractionof B, c

�

Number of states,¹

��, for W

Number of states,¹

��, for B

Fermi energy,E�(Ry)

Total energyE��(Ry)

0.001 4.959 2.555 0.273 !32.2240.01 4.975 2.550 0.272 !32.3200.091 5.186 2.518 0.197 !33.3840.167 5.349 2.480 0.185 !34.324

Fig. 4. Dependencies of the total energy per atom for interstitialW}B solid solutions on the lattice parameter for di!erent atomicfractions of boron. Solid solution is studied for the ordered statewith the partial occupation of only one sublattice. The left Y-axisgives energies for c

�"0.001 and the right one gives energies for

c�"0.01. Boxes 1 and 2 show the region with the positive

curving in the energy plot for c�"0.001. Arrows show the

direction of shifts of this curving with a concentration change.

one of the interstitial sites sublattices in tungstenand (b) to gain knowledge about the bonding ten-dencies for boron atoms in tungsten media.At the "rst step of calculations we investigated

the structure where boron occupies with small con-centrations only the corners of one cubic primitivecell of octahedral interstitial positions. At eachstage of calculations, i.e. at all the studied concen-trations the lattice parameter was varied to obtainthe minimal total energy. The increase of boronconcentration has an opposite e!ect on the partialtotal energies: the tungsten energy decreases whilethe boron energy increases. Table 1 illustrates thesame tendency: the total number of states at theFermi energy for W increases while for B it de-creases with the growth of boron concentration. Atthe same time the total energy per atom for diluteW}B interstitial solid solutions decreases as well asFermi energy. The partial numbers of s- and p-states for W show the stable tendency to increasewith rising boron concentration while the numberof s- and p-states for B decrease. This is not the casefor the behavior of d-states. For W it increases,while for boron it decreases as concentration in-creases, and at concentration c

�"0.167 it starts to

increase, thus showing a nonmonotonic behavior.In Fig. 4 we present the dependence of the energyper atom on the lattice parameter for two concen-trations of extremely dilute solid solutions. Thepresented curves show nontrivial behavior. Wemay analyze these curves in terms of enthalpy be-cause the pressure}volume term in the units, whichare used for calculations, is negligibly small. Thecaving in the curve between points 1 and 2 in this"gure shows that for the lattice parameter 6.2 a.u.and for the atomic fraction of boron equal to 0.001

the tungsten-based solid solution is unstable. Thusthe two-phase mixture of dilute solid solutions withlattice parameters 6.14 and 6.26 a.u. is preferable incomparison with the one-phase state of the solidsolution. This means that although for pressureequal to zero the one-phase state with the latticeparameter a"6.14 a.u. is stable when tensile stressis applied the solid solution decomposes intoa two-phase state. When concentration of boronincreases the e!ect weakens as shown by the dashedline in Fig. 4 and vanishes at larger concentrationsof boron. From Fig. 5 it follows that the equilib-rium lattice parameter of the interstitial solid solu-tion of boron in tungsten is a nonlinear function of


Fig. 5. Dependencies of the lattice parameter (dashed line) andthe energy (solid line) of tungsten}boron alloy on the atomicfraction of boron, c

�. A solid solution is studied for the ordered

state with the partial occupation of only one sublattice.

Table 2Total energies, E

��, per cell for W}B solid solution for di!erent

boron occupations of the sites 1��

��, 2�

�� , 3�

��

and4�

�� in the "rst and second sublattices of octahedral inter-

stitial positions

Atomicfractionof B, c

�

Occupation ratio E��(Ry)

1st sublattice 2nd sublattice

1 2 3 4

0.2 0.5 0.5 0.5 0.5 !34.32730.2 1 0 0 0 !34.3463

Table 3Total energies, E

��, per cell for W}B solid solution vs. equilib-

rium Wigner}Seitz radius, R��, for di!erent boron occupations

of the sites 1��

��and 2�

�� in the "rst sublattice of octahedral

interstitial positions. In the last three lines c�"0.33

R��(a.u.)

Occupation ratio E��(Ry)

Position 1 Position 2

2.38 0 0 !32.2162.45 1 0 !37.6562.44 0.9 0.1 !37.5762.44 0.5 0.5 !37.436

atomic fraction of boron. It may be seen in Fig.2 that for extremely small boron concentration thelattice parameter has an anomalous behavior. Ananalysis of this "gure shows that the coe$cient ofconcentration dilatation of the lattice is negativefor atomic fraction of B till the boron concentrationequals 0.006, it becomes equal to zero for thiscomposition and after this it becomes positive andconcentration-independent.One of the goals of our investigation was to

study ordering tendencies in W}B solid solutions.With this purpose we studied several possibilities ofoccupations of the sites of sublattices of the oc-tahedral interstitial positions of boron atoms. Theresults of this study are given in Tables 2 and 3. InTable 2 we demonstrate how the distribution ofB atoms on the sublattices in#uences the total en-ergy of the system. The occupation ratio for a givenatomic fraction of B atoms in W

�B��

interstitialalloy shows how B atoms are distributed betweensub-sublattices. It is easy to see that the con"gura-tion corresponding to the occupation of only onesub-sublattice is preferable as compared with theequal distribution of the same amount of boronatoms between two sublattices. In these calcu-lations the third sublattice remained empty. Theenergy data presented in Table 2 correspond to theequilibrium Wigner}Seitz radius for each case.

Table 3 complements this study by investigation ofthe ordering tendencies. Here we show the in#uenceof the occupation ratio on the equilibrium Wig-ner}Seitz radius and the total energy of the cell forthe case when only one sublattice of octahedralinterstitial sites is occupied by B. In these calcu-lations B atoms were distributed on the "rst sublat-tice while the second and third ones remainedempty. The atomic fraction of B in the system wasthe same for di!erent occupation ratios. The occu-pation ratio 1 : 0 means that all B atoms occupysub-sublattice 1 from Fig. 2. This situation corres-ponds to a maximal order of B2-type for the corre-sponding atomic fraction of B. The occupationratio 0.9 : 0.1 corresponds to the partial ordering onsublattice de"ned by sites 1 and 2 in Fig. 2. Themaximal value of the long-range order parameterfor this case is �"0.8. The equal occupation ratio0.5 : 0.5 corresponds to the random distribution of


Fig. 6. Partial density of states (PDOS*in number of s-, p-,d-states per atom per Ry) for atomic fraction of boron equal to1%: (a) plot for boron; (b) plot for tungsten. An interstitial solidsolution is studied for the ordered state with the partial occupa-tion of only one sublattice.

B between sub-sublattices 1 and 2. This is the abso-lutely disordered interstitial solid solution with�"0 on the sublattice, de"ned by n

�. At the same

time this state of the alloy may be still considered asthe ordered one in the sense that two other oc-tahedral interstitial sites sublattices are absolutelyempty (n

"n

�"0).

The presented results clearly demonstrate thetendency to form the B2-phase (CsCl-type) on thesublattice of octahedral interstitial positions. ThisB2-phase consists of boron atoms, which occupythe corners of the BCC cubic cell of sublattice whilethe centers of this cell are empty. This con"gurationis favorable in comparison with others studied ifenergies are compared. The displayed plots of den-sity of states for di!erent concentration of boron(Fig. 6) show an increased concentration of bands

in the low energy region (between !0.6 and!0.8Ry) with increase of the boron concentrationin alloy. This fact indicates the formation of chem-ical bonding in W}B solid solution. An analysis ofthe partial contributions of this band shows thatthis band is formed mainly by s- and p-orbitals oftungsten and s-orbitals of boron (see Fig. 6a and b).The DOS at Fermi level, N(E

�) is an important

quantity as it is used for the estimation of theelectronic speci"c heat, the electron}phonon coup-ling constant and even for determining the vibra-tion contribution to the entropy at "nitetemperature. The numbers of N(E

�) re#ect the

trend of metallicity in this solid solution. Fig. 7 dis-plays the behavior of the total density of states inthe vicinity of the Fermi energy. It is seen thatincrease of atomic fraction of boron leads to de-crease of the number of states at the Fermi level,thus decreasing the degree of the metallic bondingin the system. At the same time for c

�"0.167 we

"nd once more the increase of N(E�). This con-

clusion is con"rmed by the results given in Table 1,which shows the nonmonotonic change of N(E

�), as

a function of boron concentration.

5. Relaxations in W GB with and without boron

The equilibrium lattice parameter for W fromour calculations is 3.244As which is in good agree-ment with experimental data 3.165As from Ref.[37]. The cohesive energy for W from ab initiocalculations is E

� "!0.632Ry/at for equilib-

rium lattice parameter which is also good whencompared with experiment (!0.637Ry/at [38]).To examine the quality of the potentials and thevalidity of the used procedure the Debye temper-ature for W was calculated with the potentialgenerated from Eq. (14). We used the same meth-odology as in Ref. [39]. The obtained value is410K which is in excellent agreement with themeasured one (392K). After that we checked theapplicability of this potential to the propertieswhere the relaxation e!ects are very important. Toaccount for the relaxations the MC method wasused in GSA. The initial temperature ¹

��(1) plays

an important role: higher values of ¹��(1) give lar-

ger jumps in the search for the minima and increase


https://www.researchgate.net/publication/243777190_Single_Crystal_Elastic_Constants_and_Calculated_Aggregate_Properties_A_Hand_Book?el=1_x_8&enrichId=rgreq-dafc12d1-5a25-436f-a2cc-617880d672c0&enrichSource=Y292ZXJQYWdlOzI0MzIyNzE5MjtBUzoxMDI3NDI4NTcwOTMxMzBAMTQwMTUwNzIxNTE2NQ==

Fig. 7. Dependencies of total density of states (in number of states per atom per Ry) on the energies normalized to the Fermi energy, E�,

in interstitial W}B solid solutions for di!erent atomic fractions of boron in the Fermi energy region. c�is the atomic fraction of boron.

A solid solution is studied for the ordered state with the partial occupation of only one sublattice.

Fig. 8. Interatomic potentials calculated from the recursionprocedure.

the chances of not pondering in a limited area of thehypersurface, obtaining then a more e$cientsweeping. Our potentials give a good convergence

for q�close to 2.6. The parameter q

�is located

around the value 2. The energy of the vacancyformation obtained in our simulations isE��

"4.23 eV. This is in good agreement withexperimental data and with the results of othercalculations [40]. We calculated also the value ofdi!usion barrier for the self-di!usion in W simula-ting the movement of the atom from the positiona/2(1 1 1) to a/2(0 0 0) where the vacancy was situ-ated. In these calculations we relaxed the lattice ateach step of the atom's movement along the di!u-sion path. The obtained value of the energy barrierwhich is the migration energy for the self-di!usionin W is equal to 1.67 eV (experimental value is1.63 eV from Refs. [41,42]).To get the W}B interatomic potential we calcu-

late the energy of an extremely dilute random solidsolution of boron in tungsten in CPA formalism.The atomic fraction of boron is taken to be equal to0.01. These calculations are performed for di!erentlattice parameters. After that, we follow stages 2}4to get the interatomic potential between W andB atoms, <

��(r). In Fig. 8 we display this potential

together with the potentials <��(r) and <

��(r) for

pure components.


Table 4Displacements of seven atoms near the relaxed �

��1 1 1� grain boundary in tungsten with and without boron atom. The data are given

for GB when the upper part of the cluster is shifted by 2.5As in the x-direction and for the nonshifted cluster. Numbers of atoms areshown in Fig. 9. All values are in As

Numbers ofatoms accordingto Fig. 9

Nonshifted Shifted

Without B in GB With B in GB Without B in GB With B in GB

1 0.111 0.425 0.330 0.2912 0.314 0.306 0.308 0.3023 0.461 0.455 0.220 0.0894 0.029 0.163 0.071 0.3035 0.102 0.065 0.110 0.1116 0.208 0.160 0.102 0.2407 0.025 0.041 0.045 0.103

Fig. 9. Positions of seven tungsten atoms in the vicinity of GBwith B (nonrelaxed case).

The potentials<�}�

(r) and<�}�(r) were applied

to simulate the relaxation of the ��1 1 1� grain

boundary in tungsten. The cluster included 432atoms for simulations in tungsten and one B atomwas then added in the interstitial position at the GBto simulate its in#uence on the properties of GB.The entire cluster was immersed in the crystal withmore than 5000 atoms. To study the in#uence ofthe slipping at the GB on the relaxation of atomswe simulated the process when the upper part of thecluster (above the GB) slides along the x-direction(from the left- to the right-hand side in Fig. 9) withrespect to the lower part (below the GB). In thisway we check the changes in the elastic "eld distri-bution in the process of gliding along the GB. In allconsidered cases the cluster was relaxed using theGSA technique without and with the B atom.To illustrate the distribution of the elastic "eld in

the vicinity of the GB we compare in Table 4 thedisplacements of di!erent atoms. The positions ofthe numerated atoms are shown in Fig. 9. Ourresults show that the elastic "eld oscillates andslowly decreases in the direction normal to GB. Asan example in Table 4 we give the displacements ofthe same atoms for the case when the shift betweenthe upper and lower parts of the cluster is equal to2.5As . This value of the shift corresponds to half ofthe translation of the lattice in the x-direction. Thedisplacement of atom 1 with B in the GB (Fig. 9)after the shift of 2.5As decreases by 30%. At thesame time the displacement of atom 2 for the sameGB does not change signi"cantly. The situation for

the pure GB changes with the shift in the X-direc-tion. The displacement of the "rst atom increasesthree times while the displacement of the secondatom changes insigni"cantly. The existence of B inGB produces the displacement of the "rst atom oftungsten (from Fig. 9) approximately four timeslarger in comparison with the pure GB. Sliding ofGB decreases this in#uence of B and even compen-sates partly the presence of B.Our simulations show that the elastic "eld pen-

etrates far fromGB and the displacements of atomsin the seventh layer are only four times smaller ascompared with the displacements of the atoms ofthe "rst layer from the GB. We show that thepresence of B on the �

��1 1 1� GB in tungsten

changes su$ciently the value of displacements of


Fig. 10. The pro"le of the energy barrier in the simulated shifti-ng process along the X-direction.

tungsten atoms at and near the GB. Boron in#uen-ces signi"cantly also the "ne structure of the latticein the sliding along the GB. The displacements ofthe atoms are de"ned not only by direct W}Binteractions, but also by indirect W}W interac-tions. Their values are attributed by the equilib-rium condition with respect to the energy of thesystem. Thus, even in the case of rather short-ranged W}B interatomic potential (in comparisonwith W}W potential) as displayed in Fig. 8, itwould be di$cult to predict ad hoc the in#uence ofthis additive on the structure. In the consideredcase the simple geometrical considerations alsoseemingly do not work, and only simulations withreliable potentials may throw some light on thereconstruction of the lattice. In this sense it is worthmentioning that we used interatomic potentials ex-tracted from ab initio calculations by an explicitinversion procedure.The di!erence between the energies per atom for

relaxed and nonrelaxed con"gurations representsthe elastic energy. This value was calculated foreach step of the shifting process. To display theobtained result we calculate the change of the elas-tic energy with respect to the elastic energy of

nonshifted state of the system (see Fig. 10). Thesame procedure was repeated with the cluster con-taining the boron atom at the grain boundary andthe analogous curve is also plotted in Fig. 10. It iseasy to see that the presence of B on the �

��1 1 1�

GB of tungsten increases su$ciently the value ofthe energy barrier. Thus boron prevents the slidingalong the GB and increases the resistance withrespect to the shift in this direction.

6. Summary

Our study of the electronic structure and thetotal energy characteristics of extremely dilute in-terstitial tungsten}boron solid solution in theframework of CPA approximations shows theability of ordering on the sublattice of interstitialoctahedral positions. The advantages of our calcu-lations lie in the use of the total and not one-electron energies, in the inclusion of charge transfere!ects in CPA, and in our refusal to use the e!ectivepair interactions (see, for example, Refs.[33,43}45]). Conclusions on phase stability arebased on the total energy calculations and are morereliable than those based on the sum of one-elec-tron energies. We properly calculated theMadelung energy. It should also be mentioned thatthe reliability of calculations based on pair interac-tions is poor. In the case when pair interactionswere derived from LMTO-CPA (see for exampleRef. [22]) they overestimate ordering tendencies.Our result on the B2 ordering tendency for theboron atoms on the interstitial sublattice may bereceived in calculations with pair interactionsqualitatively while quantitative accuracy will belost.Our atomistic simulations de"nitely show that

the elastic "eld penetrates far from the tungstenGB. We also show that the presence of B on the��1 1 1� GB in tungsten changes the value of

displacements of tungsten atoms at and near theGB. One of the main conclusions is that the boronmicroalloying signi"cantly in#uences also the "nestructure of the lattice in the sliding along the GB.This in#uence is de"ned not only by direct W}Binteractions, but also by indirect W}W interac-tions.


Acknowledgements

This work was supported by the Israel ScienceFoundation founded by the Israel Academy ofSciences and Humanities (Grant �380/97-11.7).We are greatly indebted to A.V. Ruban for thesupport during our CPA calculations. We arethankful to D.E. Ellis, G. Krasko, A. Gordon, andP. Wyder for useful discussions. S.D. is thankful to&&KAMEA'' program for "nancial support.

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Nonempirical simulations of boron interstitials in tungsten

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