UNIVERSIDADE ESTADUAL DE CAMPINAS
SISTEMA DE BIBLIOTECAS DA UNICAMP
REPOSITÓRIO DA PRODUÇÃO CIENTIFICA E INTELECTUAL DA UNICAMP
Versão do arquivo anexado / Version of attached file:
Versão do Editor / Published Version
Mais informações no site da editora / Further information on publisher's website:
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.66.104110
DOI: 10.1103/PhysRevB.66.104110
Direitos autorais / Publisher's copyright statement:
©2002
by American Physical Society. All rights reserved.
DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
CEP 13083-970 – Campinas SP Fone: (19) 3521-6493 http://www.repositorio.unicamp.br
Atomistic prediction of equilibrium vacancy concentrations in Ni
3Al
Maurice de Koning
Lawrence Livermore National Laboratory, P.O. Box 808, L-371, Livermore, California 94550
Caetano R. Miranda and Alex Antonelli
Instituto de Fı´sica Gleb Wataghin, Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, Sa˜o Paulo, Brazil
共Received 11 January 2002; revised manuscript received 29 July 2002; published 30 September 2002兲 We conduct a series of atomistic simulations to predict thermal equilibrium vacancy concentrations in ordered Ni3Al and compare the results to recent positron-annihilation spectroscopy experiments. Using a
tight-binding second-moment-approximation potential to describe the atomic interactions, we compute single point-defect formation free energies as a function of temperature using both a quasiharmonic approximation 共QHA兲 method and an ‘‘exact’’ technique based on nonequilibrium free-energy estimation 共NFE兲, which includes all anharmonic effects. The corresponding thermal equilibrium concentrations are then computed by minimizing the crystal free energy with respect to the defect concentrations within the noninteracting-defect approximation, following the canonical ensemble approach of Hagen and Finnis 关Philos. Mag. A 77, 447 共1998兲兴. It is found that the agreement between the NFE predictions for the effective formation enthalpies and entropies and experimental data is good for three near-stoichiometric compositions. The QHA results for the same compounds, however, deviate systematically and substantially from the experimental results, suggesting that the influence of anharmonicities on the formation thermodynamics of vacancies in ordered Ni3Al
com-pounds is significant.
DOI: 10.1103/PhysRevB.66.104110 PACS number共s兲: 61.72.Ji, 02.70.Ns, 65.40.Gr
I. INTRODUCTION
There is a general interest in the influence of native point defects on the physical and mechanical properties of techno-logical materials. Point defects play a central role in the mechanisms of mass transport in crystalline materials through diffusion, affecting, for instance, order-disorder phe-nomena in compounds, mechanical hardening through inter-actions with dislocations and grain boundaries, and elec-tronic properties in semiconductor materials. Accordingly, in order to understand and/or control the physical and mechani-cal properties of such materials, it is important to gather knowledge on the structure and thermodynamic properties of point defects in crystalline solids.
Transition-metal aluminides have drawn considerable at-tention, both from scientific as well as technological applica-tion viewpoints, due to their unusual high-temperature me-chanical properties. The nickel aluminide compound Ni3Al,
for instance, forms a L12 ordered structure that is respon-sible for an increasing yield stress with increasing temperature.1 Furthermore, Ni3Al-based alloys are resistant
to air oxidation due to their ability to maintain an adherent surface oxide film,2 and show remarkable high-temperature strength and creep resistance.3 Yet, the application of pure stoichiometric polycrystalline Ni3Al as a structural material
suffers from the inherent drawback that it easily undergoes brittle intergranular fracture.4 It is well documented, how-ever, that the addition of impurities共boron in particular兲 and their segregation to the grain boundaries 共GB’s兲 suppresses the brittle fracture and dramatically improves the tensile duc-tility of the polycrystal.5This effect is related to the fact that the segregation of boron increases the GB cohesion and al-lows stress concentrations to be relieved by slip in adjacent
grains rather than by intergranular fracture at the boundaries in the undoped alloy.
Given that the segregation process is largely mediated by point defects, it is important to obtain their thermal equilib-rium concentrations in order to predict the mechanical prop-erties of the doped alloy as a function of temperature and/or stress. Although much information has been inferred from experimental measurement of defect-dependent properties, there is still a great need for theoretical interpretation, usu-ally coming from atomistic-scale calculations. In the context of point defects, such calculations are usually directed at the computation of the relevant formation and/or migration en-ergetics. For Ni3Al, a number of theoretical studies based on
the use of empirical interatomic potentials6,7 and ab initio calculations8 has been carried out to determine the energet-ics, of point defect formation. Most of these are restricted to zero-temperature energetics, and finite-temperature effects are usually neglected entirely. To the best of our knowledge, only the study by Debiaggi and co-workers6considers finite-temperature effects on the formation process of vacancies in Ni3Al, computing formation entropies within the harmonic
approximation 共HA兲 using an embedded-atom model for the description of the atomic interactions.
Recent experimental data based on the measurement of vacancy concentrations using positron-annihilation spectros-copy has revealed high effective formation entropies for the formation of vacancies in near-stoichiometric Ni3Al.9 In
these situations, the vacancy on the Ni sublattice is dominant over that on the Al sublattice and the effective entropy asso-ciated with its formation was found to be considerably larger than that typically found in pure metals. While the typical value in pure metals is of the order of (1 –2)kB, the experi-ments indicate effective formation entropies for the Ni va-cancy ranging between 4kB and 7kB in Ni3Al. The
calcula-tions of Debiaggi and co-workers, however, do not predict such large values, leading to an effective formation entropy of only⬃2 kB. The fact that this prediction deviates signifi-cantly from the experimentally determined effective entro-pies is possibly related to three factors: 共1兲 the utilized embedded-atom potential fails in describing the vibrational modes correctly, 共2兲 electronic contributions to the entropy are important, or共3兲 the anharmonic effects neglected in the HA method play a significant role.
In this paper we focus on the last issue, studying the rel-evance of anharmonic effects on the vacancy-formation ther-modynamics of vacancies in Ni3Al. It has been shown
pre-viously that such effects can be significant in the excess free energies of defects such as vacancies, surfaces, and grain boundaries.10,11 In particular, in a recent study11 in which atomistic calculations based on the tight-binding second-moment approximation 共TBSMA兲 model12 for elemental copper were compared to experimental data, it was shown that anharmonic effects are indeed important for predicting thermal equilibrium vacancy concentrations. In order to in-vestigate such effects in the case of Ni3Al we evaluate the vacancy-formation thermodynamics from atomistic calcula-tions utilizing a similar TBSMA model for the interatomic interactions in Ni3Al.12 For this purpose, we compare the
results of excess vacancy free energies as obtained using a harmonic approach and a method based on nonequilibrium free-energy estimation in which all anharmonic effects have been taken into account explicitly.13–19 We make contact with recent experimental data by evaluating effective vacancy-formation parameters from the excess free-energy values determined from the atomistic simulations.
The remainder of this paper has been organized as fol-lows. In Sec. II we consider the computational methods uti-lized in this work, giving a summary of the utiuti-lized TBSMA model, the adopted definitions for the evaluation of point-defect excess free energies and the corresponding thermal equilibrium concentrations, and the utilized free-energy methods. The obtained results are discussed in Sec. III and we conclude with a summary in Sec. IV.
II. COMPUTATIONAL APPROACH A. Interatomic potential
For the description of the interatomic interactions in Ni3Al, we adopt the TBSMA model developed by Cleri and
Rosato.12Within this model, the cohesive energy of a system of N particles is written as Ecoh⫽
兺
i⫽1 N 共ER i⫹E B i兲, 共1兲in which ERi and EBi are the pairwise repulsive and many-body attractive band energies of atom i are given by
ERi⫽
兺
j A␣exp关⫺p␣共ri j/r0␣⫺1兲兴 共2兲 and EBi⫽⫺再
兺
j ␣ 2 exp关⫺2q␣共r i j/r0␣⫺1兲兴冎
1/2 , 共3兲 respectively, where the summations extend up to fifth neigh-bors. Here, ri j represents the distance between atoms i and j, ␣ andare labels for the two interacting atomic species andr0␣ is the nearest-neighbor distance in the ␣ lattice. The values of the model parameters A␣, ␣, q␣, and p␣for Ni3Al can be found in Ref. 12.
B. Formation free energies and thermal equilibrium concentrations
The procedure for the calculation of thermal equilibrium concentrations of point defects such as vacancies or intersti-tials in pure materials is relatively straightforward. Within the noninteracting-defect approximation, in which the point-defect concentration c is assumed to be so small that inter-actions between individual defects may be neglected, the thermal equilibrium concentration is obtained by minimizing the excess free energy per lattice site,
⌬G共c兲⫽cGf⫺TSconf共c兲, 共4兲
with respect to c. Here, Gf is the point-defect excess 共or
formation兲 free energy, T is the absolute temperature, and
Sconf(c) is the configurational entropy per lattice site, usually
taken to be the ideal entropy of mixing20
Sconf共c兲⫽⫺kB关c ln c⫹共1⫺c兲ln共1⫺c兲兴. 共5兲 Setting ⌬G(c)/c⫽0 then gives the well-known
expres-sion for the single point-defect thermal equilibrium concen-tration,
ceq⫽exp共⫺Gf/kBT兲. 共6兲
In compound materials the calculation of single point-defect concentrations is somewhat more complicated. For the case of ordered Ni3Al, for instance, one must consider the
L12structure, which consists of four interpenetrating
simple-cubic sublattices, three occupied by Ni atoms and one by Al atoms. Each lattice site can be occupied by either an atom appropriate for its sublattice, an atom of the opposite type
共an antisite defect兲 or a vacancy. The thermal equilibrium
concentrations should then be determined by minimizing the free energy with respect to the concentrations of the four point defects. Depending on the statistical ensemble consid-ered, this minimization can be performed in different ways,21 which, of course, should lead to the same results in the end. In the following we briefly summarize the canonical en-semble approach reported by Hagen and Finnis22 and which was used in the present work.
Consider the Ni3Al L12 lattice, in which we associate
each site with either the Ni or Al sublattice. The alloy may deviate slightly from stoichiometry and actually have a com-position NixAl1⫺x where, within the noninteracting-defect
approximation, it is assumed that x is within a few percent of 75%. Each site of each sublattice can be occupied by either its own atom, the other atomic species or a vacancy, so that six species define the concentrations on both sublattices: cNi
de KONING, MIRANDA, AND ANTONELLI PHYSICAL REVIEW B 66, 104110 共2002兲
represents the concentration of Ni atoms on the Ni sublattice,
cAl represents the concentration of Al atoms on the Al
sub-lattice, cNiV represents the concentration of vacancies on the Ni sublattice, cAlV represents the concentration of vacancies on the Al sublattice, cNiA represents the concentration of Al atoms on the Ni sublattice, and cAlA represents the concentra-tion of Ni atoms on the Al sublattice.
The numbers of Ni and Al sublattice sites are 3N and N respectively, with N the number of conventional unit-cell blocks in the L12 structure. Within the canonical ensemble,
the numbers of Ni and Al atoms are held fixed, while the total number of lattice sites in the system is allowed to vary due to the creation/annihilation of vacancies or varying sto-ichiometry. In addition, the concentrations should satisfy the constraints cNi⫹cNi V⫹c Ni A⫽1, cAl⫹cAl V⫹c Al A⫽1, 共7兲 and N共3 cNi⫹cAl A兲⫽n Ni, N共cAl⫹3 cNi A兲⫽n Al, 共8兲
which, respectively, represent the sum rules for the defect concentrations on both sublattices and the constraints for the fixed the numbers of Ni and Al atoms.
Within this scheme, the total free energy of a system con-taining 3N Ni sublattice sites and N Al sublattice sites can be written as22 G⫽3N共cNiGNi⫹cNi A GNi A⫹c Ni V GNi V兲⫹N共c AlGAl⫹cAl A GAl A ⫹cAl VG Al V兲⫺TS conf, 共9兲
where GNi(GAl) is the ‘‘cohesive free energy’’ of a Ni共Al兲
atom on its own sublattice in the defect-free stoichiometric alloy, GNi V , GAl V , GNi A , and GAl A
are the respective point-defect formation free energies, and Sconf is the
configura-tional entropy. The cohesive free energies GNi and GAl,
however, are not uniquely defined in an alloy. In an elemen-tal material there is no such problem and excess defect quan-tities can be measured with respect to the well-defined free energy per atom in the perfect crystal. In a compound such as Ni3Al, however, the choice of reference states is ambiguous,
given that the division of the total free energy among the different atomic species is arbitrary.22 Yet, as shown in Ref. 22, the physically relevant 共measurable兲 quantities such as thermal equilibrium concentrations or excess energies of stoichiometry-conserving defects remain independent of the particular choice of reference states. In the present work, we use cohesive free-energy definitions based on the specific allocation of energy to Ni and Al atoms within the TBSMA model. In this manner, the excess free energy of, for ex-ample, the Ni vacancy is given by22
GNiV⫽GNi共V兲⫺G0⫹GNi, 共10兲
where GNi(V) is the free energy of a cell containing 4N
lattice sites and a single Ni vacancy in an otherwise defect-free L12 structure, G0 is the free energy of the defect-free
stoichiometric structure with the same number of lattice sites, and GNiis the corresponding cohesive free energy of a
Ni atom, as prescribed by the TBSMA model. Similarly,
GAlV , GNiA , and GAlA are defined as
GAlV⫽GAl共V兲⫺G0⫹GAl, 共11兲
GNiA⫽GNi共A兲⫺G0⫹GNi, 共12兲
and
GAlA⫽GAl共A兲⫺G0⫹GAl, 共13兲
where GNi(A)„GAl(A)… represents the free energy of a cell
containing 4N lattice sites, and in which a single Ni 共Al兲 atom has been replaced by an Al 共Ni兲 atom in an otherwise defect-free structure.
After obtaining the cohesive and point-defect formation free energies, the equilibrium point-defect concentrations are determined by minimizing the total free energy in Eq. 共9兲 subject to the four constraints in Eqs.共7兲 and 共8兲. As detailed in Ref. 22, this is accomplished by introducing four Lagrange multipliers, two of which are the chemical poten-tials of the Ni and Al species. For a given alloy composition NixAl1⫺x, the solution of the corresponding equations then
gives the six equilibrium concentrations 共four point defects and the concentrations of Ni and Al on their respective sub-lattices兲, the number of lattice sites N 共in terms of the abso-lute numbers of Ni and Al atoms in the system兲, and the chemical potentials NiandAl of both atomic species.
C. Free-energy techniques
In order to predict the thermal equilibrium concentrations of point defects we must determine the formation free ener-gies defined in Eqs. 共10兲–共13兲. A straightforward approach for this purpose is the application of the HA.23 Within this scheme, the free energy of a system of particles is computed from a quadratic expansion of the potential energy about their equilibrium positions. In this view, the interactions in the system are described in terms of a collection of springs, each of which describes the coupling between the degrees of freedom of a given pair of particles. The coefficients in the harmonic expansion represent the stiffnesses of the springs and constitute the dynamical matrix of the system. Its eigen-values and eigenvectors describe the phonon modes of the structure under consideration. In the high-temperature classi-cal limit, the total harmonic free energy of the system is then given by F⫽U0⫹kBT
冕
0 ⬁ d g共兲ln冉
ប kBT冊
, 共14兲where U0 is the total potential energy of the equilibrium configuration and g() is the total vibrational density of states, defined as
g共兲⫽
兺
i ␦共⫺i兲. 共15兲 Here, the summation runs over the eigenfrequencies i of the dynamical matrix.
Using the total vibrational density of states, Eq.共15兲, one can compute the harmonic free energies of the perfect and defected cells involved in the definitions in Eqs.共10兲–共13兲. However, in order to compute the individual cohesive free energies GNiand GAlin the perfect L12 structure, we should separate the vibrational contributions due to the Ni and Al species. For this purpose we should determine the local vi-brational properties of both atomic species. The vivi-brational contribution due to particle k in a system is described in terms of its local density of states24
gk共兲⫽
兺
i ␦共⫺i兲␣⫽1
兺
3
兩
具
k␣兩i典
兩2. 共16兲Here兩i
典
is the ith eigenvector of the dynamical matrix of the entire system and兩k␣典
represents a local coordinate␣ 共e.g.,x, y or z) describing the position of particle k. Of course, the
total density of states of the system is recovered by summing the local densities of states of all particles in the system:
g共兲⫽
兺
kgk共兲. 共17兲 Using the definition in Eq.共16兲 for the local density of states, the cohesive free energies GNiand GAlare given by
GNi⫽u0,Ni⫹kBT
冕
0 ⬁ d gNi共兲ln冉
ប kBT冊
共18兲 and GAl⫽u0,Al⫹kBT冕
0 ⬁ d gAl共兲ln冉
ប kBT冊
, 共19兲 where gNi() „gAl()… describes the local density of statesof a Ni共Al兲 atom in the defect-free stoichiometric structure, and u0,Ni (u0,Al) is the corresponding equilibrium
potential-energy contribution of a Ni 共Al兲, as given by the TBSMA model.
This framework provides a relatively simple recipe for the computation of the formation free energies of the four spe-cies of point defects. First, one creates the computational cells with the perfect crystal and relevant defect configura-tions and relaxes the atomic coordinates to locate their equi-librium positions and potential energies. Next, one evaluates the corresponding dynamical matrices and diagonalizes them to identify the eigenvectors and eigenvalues. From these, one constructs the appropriate vibrational density-of-states func-tions and computes the free energies using Eqs. 共14兲, 共18兲, and 共19兲. In addition, finite-temperature effects associated with thermal expansion can be taken into account within the quasiharmonic approximation共QHA兲 framework,23in which the equilibrium positions and dynamical matrices of the vari-ous structures are computed at the volume appropriate to the temperature under consideration.
A potentially serious drawback of the共quasi兲harmonic ap-proximation, however, is the fact that it ignores effects re-lated to the anharmonic motion of the particles. While such effects are usually minor in perfect solids up to relatively elevated temperatures, it has been shown that they may no longer be negligible in the presence of defects, even at lower temperatures.10,11 In order to investigate the possible influ-ence of such effects on the formation thermodynamics of the two vacancies in the L12 structure, we also compute their excess free energies using an approach that includes all an-harmonic effects. The method is based on the techniques of nonequilibrium free-energy estimation共NFE兲,13–18and mea-sures the work required to reversibly remove a Ni 共or Al兲 atom from the perfect lattice using finite-temperature molecular-dynamics共MD兲 simulations. This is accomplished by introducing a coupling parameter into the potential-energy function that controls the strength of the interaction between the specified atom and all of its neighbors. By switching the parameter from ⫽1 共perfect crystal兲 to
⫽0 共system with vacancy兲 and measuring the associated
re-versible work, one determines the free-energy difference be-tween the cell containing the vacant lattice site and that of the perfect lattice structure including all anharmonic effects. The details of this procedure have been described elsewhere.19In order to compute the formation free energies defined in Eqs. 共10兲 and 共11兲, we must also determine the cohesive free energies of both atomic species in the perfect crystal. The energetic parts of GNi and GAl are determined easily from equilibrium MD simulations by measuring the average potential energies of Ni and Al atoms in the perfect crystal, as prescribed by the TBSMA model. The correspond-ing entropic contributions are computed from the Fourier transform of the velocity-velocity autocorrelation functions following the method described by Ravelo and co-workers.25
D. Calculations
We utilized computational cells containing 500 and 864 atoms subject to standard periodic boundary conditions. The 864-atom cells were utilized to verify the magnitude of finite-size effects on the results of the QHA calculations. It was found that the results obtained with the larger cells were essentially identical to those obtained with the smaller cells over the entire temperature range. Accordingly, the finite-temperature NFE calculations were carried out only for the 500-atom cells.
As a first step, we determined the equilibrium volume of the perfect L12 structure as a function of temperature for
zero external pressure. For this purpose we carried out a series of equilibrium MD simulations within the isobaric-isothermal 共NPT兲 ensemble19 for different temperatures be-tween 200 and 1400 K.
For the QHA calculations we created four cells for each temperature, one for each type of point defect, using the equilibrium volumes determined from the MD simulations of the perfect L12 configuration. After relaxing the atomic de-grees of freedom at constant volume using a conjugate-gradient minimization scheme, we determined the respective dynamical matrices and corresponding eigenvectors/
de KONING, MIRANDA, AND ANTONELLI PHYSICAL REVIEW B 66, 104110 共2002兲
eigenvalues. The resulting data were then used in the expres-sions of the harmonic approximation to compute the point-defect formation free energies as a function of temperature.
The anharmonic NFE calculations were limited to the ex-cess free energies of the two vacancies. We did not compute them for the antisite defects because their formation energies are relatively small while the thermal noise in the reversible work determination is relatively large. The formation free energies of the vacancies were determined as a function of temperature using the same cells 共same volume兲 as those employed in the QHA calculations. For each temperature we applied the nonequilibrium switching method,19 reversibly introducing a vacancy at constant volume and temperature during nonequilibrium simulations of a duration of 4 ps each. Systematic errors related to the intrinsic irreversibility of the process were found to be negligibly small for these runs, while the statistical errors were determined from 10 indepen-dent switching replicas for each temperature. In order to compute the cohesive free energies GNi and GAl, we
com-puted the average enthalpies and velocity-velocity autocorre-lation functions for both atomic species from a series of constant-volume equilibrium MD simulations of the defect-free crystal. Each simulation was carried out at the volume appropriate to the temperature under consideration, spanning a total time interval of 0.5 ns.
III. RESULTS AND DISCUSSION
Figures 1 and 2 show the formation free energies as a function of temperature of single vacancies on the Ni and Al
sublattices in an otherwise defect-free cell for the TBSMA potential. The triangular symbols represent the data obtained using the QHA method. The QHA formation free energies decrease essentially linearly with temperature, giving rise to temperature-independent formation enthalpies HQHA
V
and en-tropies SQHAV in the relation
GQHAV 共T兲⫽HQHAV ⫺TSQHAV . 共20兲 Table I shows the values of the QHA formation enthalpies and entropies for both vacancies. The circular symbols with error bars represent the results of the NFE calculations. Due to the explicit anharmonicity,11both the formation enthalpies and entropies depend explicitly on temperature, approaching the QHA values for low temperatures. In the present case we found the formation free energies obtained with both meth-ods to become essentially identical around a temperature of
T0⫽100 K. Assuming that the formation enthalpy increases
linearly with increasing temperature beyond T0, one can fit
the NFE formation free energies to the expression
GNFEV 共T兲⫽HQHAV ⫹␣kB共T⫺T0兲⫺TSQHA V ⫺␣ kBT ln
冉
T T0冊
, 共21兲FIG. 1. Excess free energy as a function of temperature of a vacancy on the Ni sublattice in perfectly stochiometric Ni3Al.
Tri-angular symbols represent results obtained using the QHA method. Straight lines are results of linear regression to data. Circles repre-sent the data obtained using the NFE method. Error bars were com-puted from ten independent runs. The curved line is the result of fitting the data to Eq.共21兲.
FIG. 2. Excess free energy as a function of temperature of a vacancy on the Al sublattice in perfectly stochiometric Ni3Al.
Tri-angular symbols represent results obtained using the QHA method. Straight lines are results of linear regression to data. Circles repre-sent the data obtained using the NFE method. Error bars were com-puted from ten independent runs. The curved line is the result of fitting the data to Eq.共21兲.
TABLE I. QHA formation enthalpies HQHA
V
共in eV兲 and
entro-pies SQHA
V 共in k
B) for both vacancies.
HQHA V (Ni) SQHA V (Ni) HQHA V (Al) SQHA V (Al) 1.94 1.48 2.04 1.45
which satisfies the thermodynamic relation dSNFEV /dT
⫽(1/T)dHNFE
V
/dT and in which␣ is the only adjustable pa-rameter. The results are shown as the curved lines in Figs. 1 and 2, in which␣ has values of 1.81 and 0.77, respectively. Further inspection of the particular choice of T0⫽100 K in the fitting procedure has shown that small variations in
T0 (⌬T0⫽⫾50 K) lead to deviations smaller than 0.5%
over the entire temperature range and have no significant effect on the results reported below.
Figures 3 and 4 show the difference between the NFE and QHA formation enthalpies and entropies as a function of
temperature. The QHA technique is seen to give substantially lower values for the formation entropies as compared to those predicted by NFE. The fact that the calculations with both methods were carried out using computational cells with the same volume indicates that anharmonic motion of the atoms, which is enhanced due to the loss of local inver-sion symmetry around the vacancy, is not negligible and gives a significant contribution to the formation entropies.
Figure 5 shows the results of the antisite formation free energies for the two types of antisite defects as a function of temperature, computed with the QHA method. Instead of the bare formation free energies defined in Eqs. 共12兲 and 共13兲, the plot shows the free-energy differences GNiA⫺GNi and
GAl
A⫺G
Al, which appear explicitly in the equations for the
crystal free-energy minimization from Ref. 22. We did not attempt to determine them using the anharmonic method, given that the antisite formation energies are small and the thermal noise in the NFE calculations for point defects is relatively large. The free-energy difference for the antisite defect on the Al sublattice is significantly lower than those of all other point defects, of the order of only⬃0.12 eV, and is approximately independent of temperature. The free-energy difference for the Ni antisite is somewhat higher, but still considerably lower than those of the vacancies. Its formation enthalpy and entropy are, respectively, Hf⫽0.77 eV and Sf ⫽1.63kB.
Having computed the temperature dependence of the for-mation free energies of the four point defects, we are now in a position to predict their thermal equilibrium concentrations as a function of temperature by minimizing the total crystal free energy. Given that the minimization is done within the noninteracting defect approximation, the determination of equilibrium concentrations is valid only for compositions close to ideal stoichiometry.
FIG. 3. Difference between the NFE and QHA formation enthal-pies for both vacancies as a function of temperature.
FIG. 4. Difference between the NFE and QHA formation entro-pies for both vacancies as a function of temperature.
FIG. 5. QHA excess free energies as a function of temperature of both antisite defects in perfectly stochiometric Ni3Al. Triangular
symbols represent results for the antisite defect on the Ni sublattice; circles show data for the antisite defect on the Al sublattice.
de KONING, MIRANDA, AND ANTONELLI PHYSICAL REVIEW B 66, 104110 共2002兲
Within this assumption, the present atomistic results can be compared to the recent experimental data obtained by Badura-Gergen and Schaefer.9 They carried out a series of positron-annihilation spectroscopy measurements in which they measured the temperature dependence of the formation of thermal vacancies in three nearly stoichiometric composi-tions of Ni3Al: Ni74.1Al25.9, Ni75.2Al24.8, and Ni76.5Al23.5.
Measuring the positron trapping rate for the three composi-tions, they estimated the thermal vacancy concentration cVas a function of temperature and determined the corresponding effective formation enthalpies HVFand entropies SVF by fitting the results to the Arrhenius law,
ln cV⫽ SVF kB ⫺ HV F kBT . 共22兲
Utilizing the atomistic single point-defect free energies and the noninteracting-defect approximation we can determine these effective formation parameters for the TBSMA model and compare the results to the experimental positron-annihilation data.
Figure 6 shows the logarithm of the thermal equilibrium concentrations of the four point defects within the TBSMA model as a function of the inverse temperature for the Ni75.2Al24.8 compound. They were obtained by substituting
the anharmonic NFE excess free energies of the vacancies and the QHA formation free energies of the antisite defects into the equations for minimizing the crystal free energy. As expected, due to their low formation free energies, the anti-site defects are dominant by far, with concentrations between 4 and 15 orders of magnitude higher than those of the vacan-cies across the shown temperature interval. Considering the
vacancies, the atomistic results indicate that the vacancy on the Ni sublattice is predominant, with a concentration of about two orders of magnitude larger than that of Al vacan-cies. The temperature dependence of the vacancy concentra-tions follows, to a very good approximation, an Arrhenius law. Accordingly, by fitting the equilibrium thermal concen-trations determined from the atomistic calculations to Eq.
共22兲 one obtains estimates for the effective
vacancy-formation parameters measured in the experiments.
Table II compares the effective vacancy-formation param-eters determined from the atomistic calculations and those obtained in the positron-annihilation experiments of Badura-Gergen and Schaefer. For each of the three compositions, we report two sets of results for the TBSMA model, one ob-tained using only QHA formation free energies in the calcu-lation of the equilibrium concentrations, and one obtained with the fully harmonic NFE results for the vacancy excess free energies.
For the compound Ni74.1Al25.9, the effective formation
enthalpies predicted by the TBSMA model and the noninteracting-defect approximation agree quite well with the experimentally measured value, both for the QHA and NFE calculations. For the formation entropy, however, there is a significant discrepancy between the two methods. The formation entropy SVF⫽0.16kB predicted from the QHA cal-culations is much lower than the value SVF⫽3.91kB obtained using the fully anharmonic NFE excess vacancy-free ener-gies. The latter result, however, agrees very well with the effective formation entropy SVF⫽3.94kB measured in the ex-periments. Considering the other two compositions, the agreement between the NFE calculations and the experimen-tal data for Ni75.2Al24.8 and Ni76.5Al23.5 remains quite good.
The NFE results reproduce the trend of increasing formation enthalpy and entropy with increasing Ni concentration and give effective formation enthalpies and entropies that, con-sidering the error bars in Figs. 1 and 2, are in reasonable agreement with the experimentally measured values. On the other hand, the QHA results predict effective formation en-tropies that are systematically and significantly lower than those predicted in the NFE calculations and those measured in the positron-annihilation spectroscopy experiments. While the discrepancy between the NFE formation entropies and
FIG. 6. Arrhenius plot of the thermal equilibrium concentrations of the four types of point defects predicted from the atomistic ex-cess free energies. The results were obtained using the NFE forma-tion free energies for the two vacancies and the QHA values for the antisite defects.
TABLE II. Effective vacancy formation enthalpies HV F
共in eV兲
and entropies SV F共in k
B) for compositions with Ni concentrations cV
of 74.1%, 75.2%, and 76.5%. Two sets of data are reported for the atomistic calculations. The set marked QHA was obtained using only QHA formation free energies in the calculation of the equilib-rium concentrations. The results marked with NFE were those ob-tained using the anharmonic NFE excess free energies for the va-cancies. The experimental results reported by Badura-Gergen and Schaefer, Ref. 9, are also shown.
QHA NFE Experiment
CNi HV F SV F HV F SV F HV F SV F 74.1 1.75 0.16 1.86 3.91 1.65⫾0.08 3.94 75.2 1.93 1.92 2.03 5.67 1.81⫾0.08 4.86 76.5 1.96 2.04 2.07 5.79 2.01⫾0.08 7.0
the experimental values is ⬃17% or less, the QHA values are at least a factor of 2–3 smaller than the positron-annihilation data. This suggests that explicit anharmonic ef-fects play a significant role in the determination of point-defect equilibrium concentrations in Ni3Al from atomistic
calculations.
IV. SUMMARY
We have carried out a series of atomistic calculations to predict thermal equilibrium concentrations of point defects in
L12Ni3Al as described by the TBSMA potential due to Cleri
and Rosato.12 The equilibrium concentrations were deter-mined by minimizing the excess free energy of the crystal with respect to the point-defect concentrations within the noninteracting defect approximation, using the canonical en-semble approach model reported by Hagen and Finnis.22For this purpose we computed the excess free energies of the monovacancies and antisite defects on the Ni and Al sublat-tices as a function of temperature. In order to evaluate the influence of anharmonic effects, we computed them using both QHA method and an exact technique based on NFE, which includes all anharmonic effects. The atomistic point-defect excess free energies were then used in the noninter-acting defect equations to predict the vacancy concentrations as a function of temperature in the three nearly stoichio-metric compositions Ni74.1Al25.9, Ni75.2Al24.8, and Ni76.5Al23.5, for which recent experimental data based on
positron-annihilation spectroscopy is available.9 The agree-ment between the NFE results and experiagree-ments is found to be quite good for all three compositions. Aside from cor-rectly reproducing the trend of increasing vacancy-formation enthalpy and entropy with increasing Ni content, the NFE results are in reasonable quantitative agreement with the positron-annihilation experiments. The QHA results, how-ever, deviate significantly from both the NFE and experimen-tal results. In particular, the effective formation entropies predicted by QHA are at least a factor of 2–3 smaller than the NFE results and the positron-annihilation data. This sug-gests that the effects of anharmonicities on the formation thermodynamics of vacancies in Ni3Al are significant, and
that they should be considered in a quantitative prediction of point-defect thermal equilibrium concentrations from atom-istic calculations.
ACKNOWLEDGMENTS
This work was performed under the auspices of the US Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48. We thank A. Caro and S. Debiaggi for helpful discussions. M.K. acknowledges partial support from the Brazilian agency FAPESP under Contract No. 97/ 14290-6. C.R.M. and A.A. gratefully acknowledge support from the Brazilian funding agencies FAPESP, CNPq, and FAEP.
1P.H. Thorton, R.G. Davis, and T.L. Johnson, Metall. Trans. 1, 207
共1970兲.
2E.A. Aitken, in Intermetallic Compounds, edited by J.H.
West-brook共Wiley, New York, 1967兲, pp. 491–515.
3G. Sauthoff, in Diffusion in Ordered Alloys, edited by B. Fultz,
R.W. Cahn, and D. Gupta, TMS, EMPMD Monograph Series Vol. 3共TMS, Warrendale, 1993兲, p. 205.
4C.T. Liu, Scr. Metall. Mater. 15, 1231共1991兲.
5K. Aoki and O. Izumi, Nippon Kinzoku Gakkaishi 43, 1190
共1979兲.
6S.B. Debiaggi, P.M. Decorte, and A.M. Monti, Phys. Status Solidi
B 195, 37共1996兲.
7S.M. Foiles and M.S. Daw, J. Mater. Res. 2, 14共1987兲. 8C.L. Fu and G.S. Painter, Acta Mater. 45, 481共1997兲.
9K. Badura-Gergen and H.E. Schaefer, Phys. Rev. B 56, 3032
共1997兲.
10
S.M. Foiles, Phys. Rev. B 49, 14 930共1994兲.
11N. Sandberg and G. Grimvall, Phys. Rev. B 63, 184109共2001兲. 12F. Cleri and V. Rosato, Phys. Rev. B 48, 22共1993兲.
13M. Watanabe and W.P. Reinhardt, Phys. Rev. Lett. 65, 3301
共1990兲.
14W.P. Reinhardt and J.E. Hunter, Jr., J. Chem. Phys. 97, 1599
共1992兲.
15J.E. Hunter, Jr., W.P. Reinhardt, and T.F. Davis, J. Chem. Phys. 99, 6856共1993兲.
16C. Jarzynski, Phys. Rev. Lett. 78, 2690共1997兲; Phys. Rev. E 56,
5018共1997兲.
17M. de Koning, A. Antonelli, and S. Yip, Phys. Rev. Lett. 83, 3973
共1999兲; J. Chem. Phys. 115, 11 025 共2001兲.
18M. de Koning, W. Cai, A. Antonelli, and S. Yip, Comput. Sci.
Eng. 2, 88共2000兲.
19M. de Koning and A. Antonelli, Phys. Rev. B 55, 735共1997兲. 20S.M. Foiles and M.S. Daw, J. Mater. Res. 2, 5共1987兲. 21Y. Mishin and C. Herzig, Acta Mater. 48, 589共2000兲. 22M. Hagen and M.W. Finnis, Philos. Mag. A 77, 447共1998兲. 23A.A. Maradudin, E.W. Montroll, G.H. Weiss, and I.P. Ipatova,
Theory of Lattice Dynamics in the Harmonic Approximation,
2nd ed.共Academic, New York, 1971兲.
24J. Li and S. Yip, Phys. Rev. B 56, 3524共1997兲.
25R. Ravelo, J. Aguilar, M. Baskes, J.E. Angelo, B. Fultz, and B.L.
Holian, Phys. Rev. B 57, 862共1998兲.
de KONING, MIRANDA, AND ANTONELLI PHYSICAL REVIEW B 66, 104110 共2002兲
