- Email: [email protected]
Issues in the ab-initio assessment of hcp transition metals self-diffusion
Physica B 407 (2012) 3298–3300
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Issues in the ab-initio assessment of hcp transition metals self-diffusion R.C. Pasianot a,b,c,n, R.A. Pe´rez a,b,c a b c
Depto. Materiales, CNEA-CAC, Avda. Gral. Paz 1499, 1650 San Martı´n, Argentina CONICET, Avda. Rivadavia 1917, 1033 Buenos Aires, Argentina Instituto Sabato, UNSAM/CNEA, Avda. Gral. Paz 1499, 1650 San Martı´n, Argentina
a r t i c l e i n f o
a b s t r a c t
Available online 17 December 2011
Electronic structure techniques, as embodied in the SIESTA code, are presently used for assessing selfdiffusion in the hcp transition metals Zr and Ti. Several issues pertaining to this apparently hard case and not routinely found in the literature, are touched upon, a partial list including: (i) the suitability of available pseudopotentials, (ii) the need to employ fine space grids for the numerical integrations, (iii) the need of a rather large basis set, (iv) the reliability of the simulation cell size and boundary conditions, etc. All of which affect the precision of the magnitudes to be evaluated, namely, in the framework of transition state theory and assuming a standard vacancy mechanism, formation energy and entropy, migration energies, and attempt frequencies. & 2011 Elsevier B.V. All rights reserved.
Keywords: Ab-initio Diffusion Hcp metals Zr Ti
1. Introduction Since its original introduction in the 1960s [1,2], density functional theory (DFT) has evolved in a powerful tool currently impacting not only on fundamental solid state physics, but also on traditional materials science. This is possible thanks to the ever increasing, affordable, computing power of nowadays, and also due to the availability on the Internet of sophisticated computing codes, either free or for moderate prices [3–7]. Among the first applications in the latter field and perhaps still the most common, is the calculation of energy and structure of relatively small ( 10 atoms) crystalline cells, found in nature or hypothetical, useful e.g. for phase diagram predictions. The next stage in complexity involves the evaluation of microstructural defects (typically point defects) which is computationally more expensive ( 100 atoms), and in many instances the most reliable way to access fundamental defect properties that are unavailable or hindered to experimental observation; these type of studies are bringing new understanding to a rather aged research field as radiation damage, e.g. Refs. [8–11], to cite but a few. With the exception of a couple of pioneering works [12,13], the application of ab-initio techniques to diffusion problems is of more recent date. Within the framework of transition state theory (TST) [14], this involves not only the energetic aspects of defects, but also their vibrational properties. To the authors’ knowledge however, to date only two of these studies can be considered as
n Corresponding author at: Depto. Materiales, CNEA-CAC, Avda. Gral. Paz 1499, 1650 San Martı´n, Argentina. E-mail address: [email protected] (R.C. Pasianot).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.12.092
fully self-consistent, ab-initio, based. One of them, dealing with self-diffusion in Al [15], is the corollary of several investigations devoted to that metal; the VASP code [7] and Vineyard’s TST [14] were employed; agreement with experiment was claimed to be very good. The other one deals with self- and substitutional impurity (Mo,W,Ta,Hf) diffusion in Fe [16]; VASP and TST were also used and overall agreement with experiment resulted within a factor of five (though comparable to the data scatter in the paramagnetic phase). In a previous work [17] devoted to effects of Fe on the diffusion in hcp Zr, we were able to identify a candidate mechanism for the strong self-diffusion enhancement caused by the presence of the impurity; a combination of SIESTA [4] and WIEN2k [3] ab-initio codes was used, and the analysis was supported only on energy calculations. Thus, as a preliminary of a more complete description including dynamic factors, an investigation of the relatively simpler, vacancy-mediated, selfdiffusion in Zr, seemed an adequate vehicle to test the reliability of the approach. Such a task is here undertaken, with the necessary inclusion of Ti, an hcp transition metal in the same IVB group of the periodic chart, which is experimentally better characterized regarding material purity and diffusion behavior.
2. Methods The present electronic structure calculations are performed with SIESTA, a code based on numerical, localized, atomic orbitals, and pseudopotentials. The latter are an essential ingredient, and we found that in order to obtain lattice parameters close to experiment, ˚ c=a ¼1.593/1.588 for Zr/Ti, the GGA approach for a¼3.232/2.951 A,
R.C. Pasianot, R.A. Pe´rez / Physica B 407 (2012) 3298–3300
exchange and correlation, and inclusion of semicore states (4s2 4p6 =3s2 3p6 for Zr/Ti), were imperative; this leads to ˚ c=a ¼1.584/1.585. The orbital basis used can be a¼ 3.241/2.908 A, described as DZ with a polarized 5s state for Zr, and for Ti as SZ semicore states, together with DZ valence states including a polarized 4s one. Apparently, these characteristics lead to hard pseudopotentials, forcing to employ rather fine spatial integration grids (mesh cutoff of 450 Ry) and to take measures against the socalled ‘egg box’ effect. Otherwise, the hexagonal symmetry of the simulation cell is easily broken upon energy relaxation. With the main elements of the calculation tool set, the objective of these calculations is now described, namely, selfdiffusion through the thermally activated, random walk of vacancies. In the hcp lattice, two nearest neighbors vacancy jumps are possible, basal (B) and axial (A). The diffusion coefficient possesses thus two components, one along the c axis, DJ , contributed only by A-jumps, and the other perpendicular to it, D? , contributed by both jumps. In formulas, A
=kB T
D? ¼ 32 a2 f BX eSf =kB eQ
B
Property
=kB T
Ti
C36
C96
C36
C96
2.20 1.51 0.70 0.50 18.56 14.77
1.95 3.69 0.71 0.67 – –
1.91 1.79 0.53 0.38 11.93 24.64
2.06 2.27 0.49 0.44 – –
1e-16
Ti
Exp1 Exp2 bare corrected
1e-17
W B þ 12a2 f AX eSf =kB eQ
and
Zr
Ef Sf EAm EBm WA WB
W A, A
=kB T
W A,
W ¼ Po0i =P0 o00i ,
1e-18
ð1Þ
where f are correlation factors [18] presently set to the fcc value of 0.78, QX are activation energies, in turn made out of vacancy formation and migration terms, Q ¼ Ef þ Em ; Sf is the formation entropy, and WX are the attempt frequencies. According to Ref. [14], the latter two quantities are expressed as Sf ¼ kB lnðPoi =Po0i Þ
Table 1 Vacancy properties for Zr and Ti calculated with 36- and 96-sites simulation cells (cols. C36 and C96). Formation energy, Ef (eV); formation entropy, Sf (kB); axial (A) and basal (B) migration energy, EXm (eV), and corresponding attempt frequency, WX (THz).
ð2Þ
where o stands for the vibration eigenfrequencies of the perfect lattice, o0 are those of the defective lattice with a single vacancy, and o00 those of the defective lattice with the jumping atom at the activated (saddle point) configuration. In the product of the latter the (imaginary) frequency of the (unstable) mode leading to migration is suppressed. The expression for Sf corresponds to the high T limit, namely, above the Debye temperature, thus fully justified under practical diffusion conditions. But for the perfect lattice, the computation of the potential energy Hessian, necessary to obtain the frequency spectrum, is the most time consuming step of all the above. Use of point symmetries is therefore almost mandatory; also, this is why previous works invariably employed relatively small supercells, 32 sites in Ref. [15], 54 in Ref. [16], though larger cells might have been used for other energy calculations. Another important point to check here is that the pseudopotential must obtain a phonon dispersion diagram in reasonable agreement with experiment, exigence currently satisfied.
3. Results Two hexagonal supercell sizes are employed for computing the above mentioned quantities, 36- and 96-sites (3 3 2 and 4 4 3 repeat units respectively), keeping volume and shape fixed (to the perfect lattice values). Table 1 summarizes our results. Ef and Em figures are similar to reports in the literature, e.g. Refs. [19,20]; an important observation at this point regards the use of the so-called ‘ghost atoms’ to correct for the basis size superposition error (one ghost for the vacancy, two ghosts for the barrier calculations), that turned out to be absolutely essential for the case of Ti. A striking result is the increase of the vacancy formation entropy for Zr, that goes from a reasonable C36 value to an unexpectedly large one for C96. A browse at the extant experimental data, e.g. Ref. [21], also confirms that so large entropies are rarely to be found; the reason for such a behavior is currently unclear. Besides, those Sf figures must be corrected for constant
D (m2/s)
DJ ¼ 34c2 f AZ eSf =kB eQ
3299
1e-19
1e-20
1e-21
1e-22 1200
Tαβ 1150
1100
1050
1000
950
900
850
T (K) Fig. 1. Ti self-diffusion data; experiments are from Refs. [23] (squares) and [24] (circles). DJ calculations are labeled as ‘bare’ and ‘corrected’. See main text for full details.
(zero) pressure conditions, procedure seldom mentioned in the literature; this is effected through the relationship, Sp ¼ Sv þ bBV r , where b is the (volumetric) thermal expansion coefficient, B the bulk modulus, and Vr the vacancy relaxation volume (difference between free and fixed body surface conditions). An extra difficulty thus appears, namely, Vr for C36 cells amounts to a very large 40% of the atomic volume, consistent with the about 1 GPa stresses built on the supercell after vacancy creation (the situation is still worse for C96). In fact, direct simulation using variable volume cells leads to almost null Sf values. Interestingly, among the scanty data on ab-initio vacancy relaxation volumes, Ref. [22] reports a similar value for Zr (computed via VASP and ultrasoft pseudopotentials). An educated expectation might be 10% that would lead to a corrective entropy term of about 0:5kB , criterion here followed. Regarding the attempt frequencies, the computer resources at our disposal make it too difficult the calculation for C96 cells; two observations are however in order. Firstly, being the involved expressions similar to the ones for the entropy, it is deemed that the corresponding values are better converged than for the latter; this is because the two configurations being compared are closer in structure than the ones being compared in Sf. Secondly, any of the W’s is greater than the largest (perfect lattice) phonon frequency (about 6/9 THz for Zr/Ti); this has been observed before, though not generally recognized, and was heuristically attributed to migration entropy contributions [15]. Clearly, this points out the statistical nature of the jump as opposed to a simple mechanical event. Figs. 1 and 2 gather the Arrhenius plots for Ti and Zr respectively. The calculated lines correspond to DJ (D? is about twice
´rez / Physica B 407 (2012) 3298–3300 R.C. Pasianot, R.A. Pe
3300
1e-16
Zr
1e-17
Exp1 Exp2 bare corrected
D (m2/s)
1e-18 1e-19 1e-20
discussed in the literature, but that must be considered for an unbiased assessment of the technique capabilities. For the two cases here treated, Ti and Zr, we have obtained at best semiquantitative agreement with experiment; for Zr however, this may be reflecting the presence of phenomena that render the TST powerless. Further studies on the track of the present one are clearly needed.
Acknowledgments
1e-21 1e-22 1e-23 1200
Many thanks to Dr. M. Weissman for sharing her pseudopotentials and stimulating discussions, to Lic. Ch. Hellman for tuning the Ti basis, and to CONICET PIP 804/10 for partial funding.
Tαβ 1100
1000
900
800
References
T (K) Fig. 2. Zr self-diffusion data; experiments are from Refs. [25] (squares) and [26] (circles). Calculations same as in Fig. 1 for DJ .
as large) using the C96 values of Table 1 whenever available (‘bare’ label); the experimental data do not necessarily refer to monocrystalline samples, but rather to material purity regarding Fe content. Two data sets are reported for Ti, one Fe-free [23] ( o 1 at:ppm) and one nominally pure [24] ( 50 at. ppm). The agreement with the ‘bare’ calculation is clearly less than satisfactory, however, after applying the discussed entropy correction and incrementing the activation energy in 0.1 eV, reasonable error for the energy, the ‘corrected’ line is obtained, leading to a much improved agreement. Regarding Zr, again two sample purities are reported, 30 at:ppm [25] and 160 at:ppm [26]; the ‘bare’ calculation largely overestimates the experimental data, direct consequence of the large Sf value for the C96 cell. If the C36 value is used instead, somewhat arbitrary procedure for the time being, the ‘corrected’ line is obtained. In spite of the better match, there is still room for improvement. We note in passing that there is another set of experimental data [27] reporting Zr diffusivities orders of magnitude below those from Ref. [25]; according to the present authors however, the used samples are open to criticism, thus not shown. Matters as the latter are beyond the scope of the current article, and a more in-depth analysis of experimental data and its consistency with calculations will be given elsewhere.
4. Summary and conclusions We have shown that a self-consistent ab-initio evaluation of diffusivities is plagued with calculation details, not normally
[1] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [2] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [3] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, WIEN2K An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties, ¨ Wien, Austria, 2000. Technische Universitat [4] J.M. Soler, E. Artacho, J.D. Gale, A. Garcı´a, J. Junquera, P. Ordejo´n, D. Sa´nchezPortal, J. Phys. Condens. Matter 14 (2002) 2745. [5] X. Gonze, et al., Comput. Mater. Sci. 25 (2002) 478. [6] P. Gianozzi, et al., J. Phys.: Condens. Matter 21 (2009) 395502 (19 pp.). ¨ [7] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15. [8] C. Domain, C.S. Becquart, Phys. Rev. B 65 (2001) 024103 (14 pp.). [9] S. Han, L.A. Zepeda-Ruiz, G.J. Ackland, R. Car, J. Srolovitz, Phys. Rev. B 66 (2002) 220101 (4 pp.). [10] C.C. Fu, F. Willaime, P. Ordejo´n, Phys. Rev. Lett. 92 (2004) 175503 (4 pp.). [11] D. Nguyen-Mahn, A.P. Horsfield, S.L. Dudarev, Phys. Rev. B 73 (2006) 20101 (4 pp.). ¨ [12] P.E. Blochl, E. Smargiassi, R. Car, D.B. Laks, W. Andreoni, S.T. Pantelides, Phys. Rev. Lett. 70 (1993) 2435. ¨ ¨ [13] W. Frank, U. Breier, C. Elsasser, M. Fahnle, Phys. Rev. Lett. 77 (1996) 518 (4 pp.). [14] G.H. Vineyard, J. Phys. Chem. Solids 3 (1957) 121. [15] M. Mantina, Y. Wang, R. Arroyave, L.Q. Chen, Z.K. Liu, C. Wolverton, Phys. Rev. Lett. 100 (2008) 215901 (4 pp.). [16] S. Huang, D.L. Worthington, M. Asta, V. Ozolins, G. Ghosh, P.K. Liaw, Acta Mater. 58 (2010) 1982. [17] R.C. Pasianot, R.A. Pe´rez, V.P. Ramunni, M. Weissmann, J. Nucl. Mater. 392 (2009) 100. [18] P.B. Ghate, Phys. Rev. 133 (1964) A1167. [19] C. Domain, A. Legris, Philos. Mag. 85 (2005) 569. [20] G. Ve´rite´, F. Willaime, C.C. Fu, Solid State Phenom. 129 (2007) 75. ¨ [21] H. Ullmaier, Atomic Defects in Metals, Landolt–Borstein Series, Group III, vol. 25, Springer, Berlin, 1991. [22] C. Domain, Ph.D. Thesis, Universite´ des Sciences et Technologies de Lille, March 2002. ¨ [23] M. Kopers, C. Herzig, M. Friesel, Y. Mishin, Acta Mater. 45 (1997) 4181. [24] C. Herzig, R. Willecke, K. Vieregge, Philos. Mag. A 63 (1991) 949. [25] J. Horva´th, F. Dyment, H. Mehrer, J. Nucl. Mater. 126 (1984) 206. [26] M. Lubbehusen, K. Vieregge, G.M. Hood, H. Mehrer, C. Herzig, J. Nucl. Mater. 182 (1991) 164. [27] G.M. Hood, H. Zou, R.J. Schultz, N. Matsura, J.A. Roy, J.A. Jackman, Defects Diffusion Forum 143–147 (1997) 49.
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Issues in the ab-initio assessment of hcp transition metals self-diffusion R.C. Pasianot a,b,c,n, R.A. Pe´rez a,b,c a b c
Depto. Materiales, CNEA-CAC, Avda. Gral. Paz 1499, 1650 San Martı´n, Argentina CONICET, Avda. Rivadavia 1917, 1033 Buenos Aires, Argentina Instituto Sabato, UNSAM/CNEA, Avda. Gral. Paz 1499, 1650 San Martı´n, Argentina
a r t i c l e i n f o
a b s t r a c t
Available online 17 December 2011
Electronic structure techniques, as embodied in the SIESTA code, are presently used for assessing selfdiffusion in the hcp transition metals Zr and Ti. Several issues pertaining to this apparently hard case and not routinely found in the literature, are touched upon, a partial list including: (i) the suitability of available pseudopotentials, (ii) the need to employ fine space grids for the numerical integrations, (iii) the need of a rather large basis set, (iv) the reliability of the simulation cell size and boundary conditions, etc. All of which affect the precision of the magnitudes to be evaluated, namely, in the framework of transition state theory and assuming a standard vacancy mechanism, formation energy and entropy, migration energies, and attempt frequencies. & 2011 Elsevier B.V. All rights reserved.
Keywords: Ab-initio Diffusion Hcp metals Zr Ti
1. Introduction Since its original introduction in the 1960s [1,2], density functional theory (DFT) has evolved in a powerful tool currently impacting not only on fundamental solid state physics, but also on traditional materials science. This is possible thanks to the ever increasing, affordable, computing power of nowadays, and also due to the availability on the Internet of sophisticated computing codes, either free or for moderate prices [3–7]. Among the first applications in the latter field and perhaps still the most common, is the calculation of energy and structure of relatively small ( 10 atoms) crystalline cells, found in nature or hypothetical, useful e.g. for phase diagram predictions. The next stage in complexity involves the evaluation of microstructural defects (typically point defects) which is computationally more expensive ( 100 atoms), and in many instances the most reliable way to access fundamental defect properties that are unavailable or hindered to experimental observation; these type of studies are bringing new understanding to a rather aged research field as radiation damage, e.g. Refs. [8–11], to cite but a few. With the exception of a couple of pioneering works [12,13], the application of ab-initio techniques to diffusion problems is of more recent date. Within the framework of transition state theory (TST) [14], this involves not only the energetic aspects of defects, but also their vibrational properties. To the authors’ knowledge however, to date only two of these studies can be considered as
n Corresponding author at: Depto. Materiales, CNEA-CAC, Avda. Gral. Paz 1499, 1650 San Martı´n, Argentina. E-mail address: [email protected] (R.C. Pasianot).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.12.092
fully self-consistent, ab-initio, based. One of them, dealing with self-diffusion in Al [15], is the corollary of several investigations devoted to that metal; the VASP code [7] and Vineyard’s TST [14] were employed; agreement with experiment was claimed to be very good. The other one deals with self- and substitutional impurity (Mo,W,Ta,Hf) diffusion in Fe [16]; VASP and TST were also used and overall agreement with experiment resulted within a factor of five (though comparable to the data scatter in the paramagnetic phase). In a previous work [17] devoted to effects of Fe on the diffusion in hcp Zr, we were able to identify a candidate mechanism for the strong self-diffusion enhancement caused by the presence of the impurity; a combination of SIESTA [4] and WIEN2k [3] ab-initio codes was used, and the analysis was supported only on energy calculations. Thus, as a preliminary of a more complete description including dynamic factors, an investigation of the relatively simpler, vacancy-mediated, selfdiffusion in Zr, seemed an adequate vehicle to test the reliability of the approach. Such a task is here undertaken, with the necessary inclusion of Ti, an hcp transition metal in the same IVB group of the periodic chart, which is experimentally better characterized regarding material purity and diffusion behavior.
2. Methods The present electronic structure calculations are performed with SIESTA, a code based on numerical, localized, atomic orbitals, and pseudopotentials. The latter are an essential ingredient, and we found that in order to obtain lattice parameters close to experiment, ˚ c=a ¼1.593/1.588 for Zr/Ti, the GGA approach for a¼3.232/2.951 A,
R.C. Pasianot, R.A. Pe´rez / Physica B 407 (2012) 3298–3300
exchange and correlation, and inclusion of semicore states (4s2 4p6 =3s2 3p6 for Zr/Ti), were imperative; this leads to ˚ c=a ¼1.584/1.585. The orbital basis used can be a¼ 3.241/2.908 A, described as DZ with a polarized 5s state for Zr, and for Ti as SZ semicore states, together with DZ valence states including a polarized 4s one. Apparently, these characteristics lead to hard pseudopotentials, forcing to employ rather fine spatial integration grids (mesh cutoff of 450 Ry) and to take measures against the socalled ‘egg box’ effect. Otherwise, the hexagonal symmetry of the simulation cell is easily broken upon energy relaxation. With the main elements of the calculation tool set, the objective of these calculations is now described, namely, selfdiffusion through the thermally activated, random walk of vacancies. In the hcp lattice, two nearest neighbors vacancy jumps are possible, basal (B) and axial (A). The diffusion coefficient possesses thus two components, one along the c axis, DJ , contributed only by A-jumps, and the other perpendicular to it, D? , contributed by both jumps. In formulas, A
=kB T
D? ¼ 32 a2 f BX eSf =kB eQ
B
Property
=kB T
Ti
C36
C96
C36
C96
2.20 1.51 0.70 0.50 18.56 14.77
1.95 3.69 0.71 0.67 – –
1.91 1.79 0.53 0.38 11.93 24.64
2.06 2.27 0.49 0.44 – –
1e-16
Ti
Exp1 Exp2 bare corrected
1e-17
W B þ 12a2 f AX eSf =kB eQ
and
Zr
Ef Sf EAm EBm WA WB
W A, A
=kB T
W A,
W ¼ Po0i =P0 o00i ,
1e-18
ð1Þ
where f are correlation factors [18] presently set to the fcc value of 0.78, QX are activation energies, in turn made out of vacancy formation and migration terms, Q ¼ Ef þ Em ; Sf is the formation entropy, and WX are the attempt frequencies. According to Ref. [14], the latter two quantities are expressed as Sf ¼ kB lnðPoi =Po0i Þ
Table 1 Vacancy properties for Zr and Ti calculated with 36- and 96-sites simulation cells (cols. C36 and C96). Formation energy, Ef (eV); formation entropy, Sf (kB); axial (A) and basal (B) migration energy, EXm (eV), and corresponding attempt frequency, WX (THz).
ð2Þ
where o stands for the vibration eigenfrequencies of the perfect lattice, o0 are those of the defective lattice with a single vacancy, and o00 those of the defective lattice with the jumping atom at the activated (saddle point) configuration. In the product of the latter the (imaginary) frequency of the (unstable) mode leading to migration is suppressed. The expression for Sf corresponds to the high T limit, namely, above the Debye temperature, thus fully justified under practical diffusion conditions. But for the perfect lattice, the computation of the potential energy Hessian, necessary to obtain the frequency spectrum, is the most time consuming step of all the above. Use of point symmetries is therefore almost mandatory; also, this is why previous works invariably employed relatively small supercells, 32 sites in Ref. [15], 54 in Ref. [16], though larger cells might have been used for other energy calculations. Another important point to check here is that the pseudopotential must obtain a phonon dispersion diagram in reasonable agreement with experiment, exigence currently satisfied.
3. Results Two hexagonal supercell sizes are employed for computing the above mentioned quantities, 36- and 96-sites (3 3 2 and 4 4 3 repeat units respectively), keeping volume and shape fixed (to the perfect lattice values). Table 1 summarizes our results. Ef and Em figures are similar to reports in the literature, e.g. Refs. [19,20]; an important observation at this point regards the use of the so-called ‘ghost atoms’ to correct for the basis size superposition error (one ghost for the vacancy, two ghosts for the barrier calculations), that turned out to be absolutely essential for the case of Ti. A striking result is the increase of the vacancy formation entropy for Zr, that goes from a reasonable C36 value to an unexpectedly large one for C96. A browse at the extant experimental data, e.g. Ref. [21], also confirms that so large entropies are rarely to be found; the reason for such a behavior is currently unclear. Besides, those Sf figures must be corrected for constant
D (m2/s)
DJ ¼ 34c2 f AZ eSf =kB eQ
3299
1e-19
1e-20
1e-21
1e-22 1200
Tαβ 1150
1100
1050
1000
950
900
850
T (K) Fig. 1. Ti self-diffusion data; experiments are from Refs. [23] (squares) and [24] (circles). DJ calculations are labeled as ‘bare’ and ‘corrected’. See main text for full details.
(zero) pressure conditions, procedure seldom mentioned in the literature; this is effected through the relationship, Sp ¼ Sv þ bBV r , where b is the (volumetric) thermal expansion coefficient, B the bulk modulus, and Vr the vacancy relaxation volume (difference between free and fixed body surface conditions). An extra difficulty thus appears, namely, Vr for C36 cells amounts to a very large 40% of the atomic volume, consistent with the about 1 GPa stresses built on the supercell after vacancy creation (the situation is still worse for C96). In fact, direct simulation using variable volume cells leads to almost null Sf values. Interestingly, among the scanty data on ab-initio vacancy relaxation volumes, Ref. [22] reports a similar value for Zr (computed via VASP and ultrasoft pseudopotentials). An educated expectation might be 10% that would lead to a corrective entropy term of about 0:5kB , criterion here followed. Regarding the attempt frequencies, the computer resources at our disposal make it too difficult the calculation for C96 cells; two observations are however in order. Firstly, being the involved expressions similar to the ones for the entropy, it is deemed that the corresponding values are better converged than for the latter; this is because the two configurations being compared are closer in structure than the ones being compared in Sf. Secondly, any of the W’s is greater than the largest (perfect lattice) phonon frequency (about 6/9 THz for Zr/Ti); this has been observed before, though not generally recognized, and was heuristically attributed to migration entropy contributions [15]. Clearly, this points out the statistical nature of the jump as opposed to a simple mechanical event. Figs. 1 and 2 gather the Arrhenius plots for Ti and Zr respectively. The calculated lines correspond to DJ (D? is about twice
´rez / Physica B 407 (2012) 3298–3300 R.C. Pasianot, R.A. Pe
3300
1e-16
Zr
1e-17
Exp1 Exp2 bare corrected
D (m2/s)
1e-18 1e-19 1e-20
discussed in the literature, but that must be considered for an unbiased assessment of the technique capabilities. For the two cases here treated, Ti and Zr, we have obtained at best semiquantitative agreement with experiment; for Zr however, this may be reflecting the presence of phenomena that render the TST powerless. Further studies on the track of the present one are clearly needed.
Acknowledgments
1e-21 1e-22 1e-23 1200
Many thanks to Dr. M. Weissman for sharing her pseudopotentials and stimulating discussions, to Lic. Ch. Hellman for tuning the Ti basis, and to CONICET PIP 804/10 for partial funding.
Tαβ 1100
1000
900
800
References
T (K) Fig. 2. Zr self-diffusion data; experiments are from Refs. [25] (squares) and [26] (circles). Calculations same as in Fig. 1 for DJ .
as large) using the C96 values of Table 1 whenever available (‘bare’ label); the experimental data do not necessarily refer to monocrystalline samples, but rather to material purity regarding Fe content. Two data sets are reported for Ti, one Fe-free [23] ( o 1 at:ppm) and one nominally pure [24] ( 50 at. ppm). The agreement with the ‘bare’ calculation is clearly less than satisfactory, however, after applying the discussed entropy correction and incrementing the activation energy in 0.1 eV, reasonable error for the energy, the ‘corrected’ line is obtained, leading to a much improved agreement. Regarding Zr, again two sample purities are reported, 30 at:ppm [25] and 160 at:ppm [26]; the ‘bare’ calculation largely overestimates the experimental data, direct consequence of the large Sf value for the C96 cell. If the C36 value is used instead, somewhat arbitrary procedure for the time being, the ‘corrected’ line is obtained. In spite of the better match, there is still room for improvement. We note in passing that there is another set of experimental data [27] reporting Zr diffusivities orders of magnitude below those from Ref. [25]; according to the present authors however, the used samples are open to criticism, thus not shown. Matters as the latter are beyond the scope of the current article, and a more in-depth analysis of experimental data and its consistency with calculations will be given elsewhere.
4. Summary and conclusions We have shown that a self-consistent ab-initio evaluation of diffusivities is plagued with calculation details, not normally
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