- Email: [email protected]
O interstitial energetics in Ti from ab initio calculations
Computational Materials Science 32 (2005) 13–19 www.elsevier.com/locate/commatsci
O interstitial energetics in Ti from ab initio calculations ~o I. Lado-Tourin a
a,b,*
, F. Tsobnang
b,1
Department of Chemistry and Materials, Universidad Europea de Madrid, (C) Tajo s/n, 28670 Villaviciosa de Odon, Madrid, Spain b Institut Superieur des Materiaux du Mans, 44, avenue F. A. Bartholdi, 72000 Le Mans, France Received 16 July 2003; received in revised form 30 October 2003; accepted 10 May 2004
Abstract Ti-porcelain systems have been used in prosthetic dentistry since 1980. The chemical bonding in this kind of systems is generally attributed to oxidic compounds. The porcelain functions as an O donator and stoichiometric and nonstoichiometric titanium oxides are formed at the interface. These oxides result in substantial distortion of the crystalline structure and failure of bonding. A useful technique for surface conditioning is the silicon-ion implantation, which results in the formation of a metal–silicon zone protecting against O diffusion. In dental materials technology, processes on the atomic scale have been investigated insufficiently or not at all. In this work, we present ab initio calculations on the energetics of an O interstitial in a Ti crystal. The aim of our study is getting new insight into the atomic scale properties of these Ti-porcelain systems. Ó 2004 Elsevier B.V. All rights reserved. PACS: 89.02; 89.80 Keywords: Porcelain; Ab initio; Density functional theory; Dental materials; Computational modeling
1. Introduction Ti/porcelain interfaces are usually found in materials for dental applications. Ti shows a good corrosion resistance as well as an excellent biocompatibility. The main problem associated with
*
Corresponding author. Address: Department of Chemistry and Materials, Universidad Europea de Madrid, (C) Tajo s/n, 28670 Villaviciosa de Odon, Madrid, Spain. Tel.: +34-912115658; fax: +34-91-6168265/6468565. E-mail addresses: [email protected] (I. LadoTouri~ no), [email protected] (F. Tsobnang). 1 Tel.: +33-24-3214001; fax: +33-24-3214039.
these interfaces is a decrease of bond strength between the porcelain and the Ti after a certain amount of wear [1]. Some thermodynamic calculations have shown that this phenomenon is caused by a high reactivity of Ti, particularly with O, which gives rise to the formation of scaly crystalline structures in the Ti-porcelain contact zone. These structures mainly consist of stoichiometric and non-stoichiometric titanium oxides [2] and are responsible of a continuing embrittlement of these interfaces [1,3–5]. It is therefore necessary to find a way to block the excessive reactivity of Ti with O. To achieve this, the implantation of Ti with Si ions seems to
0927-0256/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.05.005
14
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
be a suitable process [5,6] as the titanium silicide structures formed after the implantation process show a very high thermodynamic stability [7,8]. This stability prevents the formation of the scaly structures observed in conventional Ti/porcelain systems, which in turn gives rise to an increased bond strength between the Ti and the porcelain. Due to their complexity (existence of several phases, different kind of bonds, defects, dislocations, . . .) these interfaces have been investigated insufficiently or not at all on the atomic scale. By using molecular modeling techniques we want to get a better understanding and improve the physical and chemical properties of these materials. As a first step, in the present work we describe some calculations on the energetics of defects in Ti. Specifically, we have focused on the study of the energetics and electronic structure of an isolated O interstitial in a Ti crystal by ab initio methods. In the abundant literature on numerical calculations involving Ti–O bonding in both the bulk and especially the surface [9–14] we have found only small pieces of information regarding the energetics of O defects in Ti [15]. Nevertheless, these previous works demonstrate the capability of ab initio methods to study geometry relaxations, electronic structure and energetics (heats of formation, diffusion barriers, etc.) associated to point defects in crystals.
taken to complete convergence with respect to the size of the basis set. The main problem is the necessity of running calculations with prohibitively large basis sets in some cases. However, this problem can be avoided by using the so-called finite basis set correction [20], which corrects the results obtained at relatively low cutoff energy and k-point sampling. The only parameter that has to be known in order for this correction term to be evaluated can be expressed as dEtot =d lnEcut , where Etot is the total energy of the system and Ecut is the cutoff energy. Thus, this parameter gives a good indication of the convergence of the calculation with respect to the energy cutoff and k-point sampling. The electronic relaxation is performed by minimizing the sum of electronic eigenvalues in a fixed potential [21]. Brillouin-zone sampling is performed using the Monkhorst–Pack scheme [22]. Geometry optimization of all atomic positions is accomplished by the BFGS method, iterated until the root mean square force on each atom is less Calculations are performed in than 0.05 eV/A. periodic boundary conditions. As the main objective of this work is focused on the energetics of the isolated defects, each repeated cell contains a single O interstitial and 16 Ti atoms.
3. Results and discussion 3.1. Stability of interstitial sites
2. Method The present work was performed within the framework of density functional theory in the local density approximation for electronic exchange and correlation, the exchange and correlation energy as a function of electron density being that of Perdew and Zunger [16], as implemented in the CASTEP code [17]. The nuclei and core electrons are represented by the ultrasoft pseudopotentials generated using the optimization scheme of Vanderbilt [18]. All pseudopotentials are used in a separable Kleinman–Bylander form [19]. The valence electron pseudowave functions are expanded in terms of plane-waves whose kinetic energy is less than a chosen cutoff energy. The use of plane waves allows the calculation to be systematically
There are two possible interstitial sites for O in Ti: the octahedral site and the tetrahedral site (Fig. 1). It is known that oxygen is soluble in a-Ti with the O atoms being randomly distributed in the octahedral interstices of the hexagonally closepacked Ti lattice [23]. We have calculated the ground-state energies for O at these two sites, both with and without geometry optimization. The results for a cutoff energy of 600 eV for the non-relaxed case are shown in Table 1. Even if anything is gained from these unrelaxed models, we present them for the sake of comparison with the relaxed results. We also present in this table the embedding energy of the O atom in Ti, that is to say, the change of energy on going from perfect bulk Ti plus an
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
15
Fig. 1. O/Ti models used in the calculations. (A) Octahedral position and (B) tetrahedral position.
Table 1 Total ground-state energies and embedding energies for O in a 16-atom Ti crystal System
E (eV)
dEtot =d lnEcut (eV/atom)
Ee (eV) ðETiþO –ETi –EO Þ
Ti 16+oct. O Ti 16+tet. O
)26084.02 )26081.82
0.015 0.015
)11.09 )8.89
isolated O atom in its triplet ground-state to the Ti crystal with the O at an interstitial site. For the sake of comparison, the embedding energy calculated with two other all electron codes is
presented in Table 2 [24,25]. As one can see the absolute values of the embedding energy largely differ from one another but the three calculations show that the octahedral position is more stable in
16
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
accordance with the experimental results. The dEtot =d lnEcut values should be less than 0.01 eV/ atom for a calculation to be considered well converged. When the crystal is allowed to relax, nothing seems to happen for the octahedral case but it is worth noting the movement of the O atom after relaxing the structure in the tetrahedral case (Fig. 2). The values obtained for the relaxed cases are shown in Table 3. The tetrahedral site seems to be unstable. The oxygen atom moves down on to the plane of the Ti atoms becoming three-fold coordinated (Fig. 2). The Ti–O distance is 1.845 A (see Table 4). From this table one can note that in relaxing the atomic positions the Ti–O distances increase in the tetrahedral case and decrease a little in the octahedral case approaching the distances found in the most important titanium oxides (from in TiO2 crystals, which present 1.930 up to 1.982 A for a three-fold coordinated O atom, and 2 A Table 2 Embedding energies in eV for O in a 16-atom Ti crystal calculated with two different all-electrons codes System
Fast structure [17]
DMol [18]
Ti 16+oct. O Ti 16+tet. O
)16.05 )13.06
)10.07 )7.62
Table 3 Total ground-state energies and embedding energies for O in a 16-atom Ti crystal after relaxation of atomic positions System
E (eV)
Ti 16+oct. O )26084.66 Ti 16+tet. O )26083.02
dEtot =d lnEcut (eV/atom)
Ee (eV) ðETiþO –ETi –EO Þ
0.015 0.015
)11.73 )10.09
Table 4 Ti–O distances in angstroms in O containing Ti crystals Non-relaxed octahedral
Non-relaxed tetrahedral
Relaxed octahedral
Relaxed tetrahedral
2.066 (6)
1.803 (3) 1.749 (1)
2.045 (6)
1.845 (3) 2.169 (1)
The figures in the parentheses indicate the number of equivalent Ti–O bonds existing in the structure.
Ti2 O3 , which presents a four-fold coordinated O atom). 3.2. Electronic properties As a first step towards a complete understanding of the bonding characteristics at Ti/porcelain interfaces, the valence electron density and the
Fig. 2. Final position of an O atom initially in a tetrahedral site after relaxation of the atomic positions.
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
Mulliken charges for the O containing systems have been calculated. The electron density for different crystal planes containing O–Ti bonds is shown in Fig. 3. For the sake of comparison the results obtained for one of the polymorphs of TiO2 (rutile) are also displayed. The Mulliken charges are presented in Table 5. From these results one can see that there is a charge transfer from the Ti atoms to the O atom, which is more important for the nearest Ti atoms. In all the cases the Mulliken charge on the O atom is )1 electron. The nature
17
of the Ti–O bond is very similar to that found in titanium oxide crystals.
4. Conclusions and perspectives Our first calculations on O containing Ti crystals show that an O atom is more stable in the octahedral configuration than in the tetrahedral one. The same result is found using different types of codes, always within the framework of the
Fig. 3. Contour plots of the valence electron density on crystal planes containing Ti–O bonds. The values marked on the upper right indicate the electron number density in units of electron/bohr3 . (A) Rutile crystal, (B) octahedral position, (C) tetrahedral position, and (D) tetrahedral position after relaxation.
18
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
Fig. 3 (continued) Table 5 Mulliken charges on the O atom and its nearest neighbors Ti atoms System
qO
qTi
Non-relaxed tetrahedral
)1.066
Non-relaxed octahedral Relaxed tetrahedral
)0.941 )0.946
Relaxed octahedral
)0.943
0.200 0.333 0.143 0.150 0.132 0.145
(1.803), (1.749) (1.845), (2.169)
local-density approximation. Furthermore, the tetrahedral position seems to be unstable and after
relaxing the atomic positions the O atom becomes three-fold coordinated like in most TiO2 crystals. This may be due to the small distance between the O atom and its nearest neighbors in the tetrahedral position. The bonding characteristics are also similar to those existing in TiO2 crystals. It has to be noted that this is a preliminary study and the results are to be considered from a qualitative point of view only. The main sources of inaccuracy in the calculation are the plane-wave cutoff, the size of the cell and the Brillouin-zone sampling. Calculations with a larger cutoff energy and a more complete Brillouin-zone sampling may be necessary. The size of the repeating cell is also
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
an important question that has to be considered [15]. The distance between the O atoms in the z for all the systems while in direction is about 9 A the x and y directions this distance decreases to This may give rise to significant electronic 5 A. interactions between the O atoms. Moreover, Brillouin-zone sampling errors and mainly lattice relaxation will be affected by the size of the system. Thus, calculations on larger cells are mandatory. The question of the origin of the oxygen is also important if one wants to obtain quantitative results. If the O atom comes from molecular oxygen, one could add to the embedding energy, the dissociation energy per atom of the oxygen molecule in order to obtain the heat of solution. This energy was calculated recently for an O interstitial in the octahedral position in a 96-atom supercell [15]. The value of )6.12 eV obtained from this calculation is not very different from the value that we obtain ()6.61 eV) when subtracting the value of the experimental dissociation energy (5.12 eV) from the value calculated for the embedding energy (see Table 3). More calculations have to be performed to be able to compare results from different research groups and set up quantitative conclusions. If one considers the O atom to be donated by the oxidic porcelain, more complicated redox reactions and charge states must be considered. In conclusion, this calculation is only a first step towards a complete understanding of the phenomena that take place at porcelain/Ti interfaces. Larger models are necessary and a more exhaustive analysis of the parameters used in the calculation has to be done if one wants to get realistic results.
19
References [1] M. Walter, K. B€ oning, P.D. Reppel, J. Dent. 22 (1994) 346. [2] K. Eichner, Dtsch. Zahn€arztl. Z 23 (1968) 373. [3] M. K€ on€ onen, J. Kivilathi, J. Biomed. Mater. Res. 28 (1994) 1027. [4] H. Kimura, C. Hrong, J. Okazaki, M. Takahashi, J. Dent. Mater. J 9 (1990) 91. [5] A. Moorman, L. Wehnert, W.B. Freesmeyer, in: Proceedings of the 22nd Annual Conference European Prosphodontic Association (EPA), Turku (Finland), 1998. [6] D.K. Brice, Appl. Phys. Lett. 16 (1970) 103. [7] G. Dearnaley, Reports on Progress in Physics, Vol. XXXII, Part II, 1969. [8] G.H. Kinchin, R.S. Pease, Rep. Prog. Phys. 2 (1955) 1. [9] P.J.D. Lindan, J. Muscat, S. Bates, N.M. Harrison, M. Gillan, J. Chem. Soc. Faraday Trans. 106 (1998) 135. [10] J. Muscat, N.M. Harrison, G. Thornton, Phys. Rev. B 59 (1999) 2320. [11] J. Muscat, N.M. Harrison, G. Thornton, Phys. Rev. B 59 (1999) 15457. [12] M.A. San Miguel, C.J. Calzado, J. Fdez. Sanz, J. Phys. Chem. B 105 (2001) 1794. [13] V. Swamy, J. Muscat, J.D. Gale, N.M. Harrison, Surf. Sci. 504 (2002) 115. [14] J. Muscat, V. Swamy, N.M. Harrison, Phys. Rev. B 65 (2002) 224112. [15] R.G. Henning, D.R. Trinkle, J. Bouchet, S.G. Srinivasan, R.C. Albers, J.W. Wilkins, submitted to Phys. Rev. Lett. [16] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [17] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1045. [18] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [19] L. Kleinman, D. Bylander, Phys. Rev. Lett. 48 (1982) 1425. [20] G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169. [21] CASTEP, MSI, San Diego, 1998. [22] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [23] R.H.J. Clark, The Chemistry of Titanium and Vanadium: An Introduction to the Chemistry of the Early Transition Elements, Elsevier, 1968, p. 277. [24] Fast-Structure, MSI, San Diego, 1998. [25] DMol, MSI, San Diego, 1998.
O interstitial energetics in Ti from ab initio calculations ~o I. Lado-Tourin a
a,b,*
, F. Tsobnang
b,1
Department of Chemistry and Materials, Universidad Europea de Madrid, (C) Tajo s/n, 28670 Villaviciosa de Odon, Madrid, Spain b Institut Superieur des Materiaux du Mans, 44, avenue F. A. Bartholdi, 72000 Le Mans, France Received 16 July 2003; received in revised form 30 October 2003; accepted 10 May 2004
Abstract Ti-porcelain systems have been used in prosthetic dentistry since 1980. The chemical bonding in this kind of systems is generally attributed to oxidic compounds. The porcelain functions as an O donator and stoichiometric and nonstoichiometric titanium oxides are formed at the interface. These oxides result in substantial distortion of the crystalline structure and failure of bonding. A useful technique for surface conditioning is the silicon-ion implantation, which results in the formation of a metal–silicon zone protecting against O diffusion. In dental materials technology, processes on the atomic scale have been investigated insufficiently or not at all. In this work, we present ab initio calculations on the energetics of an O interstitial in a Ti crystal. The aim of our study is getting new insight into the atomic scale properties of these Ti-porcelain systems. Ó 2004 Elsevier B.V. All rights reserved. PACS: 89.02; 89.80 Keywords: Porcelain; Ab initio; Density functional theory; Dental materials; Computational modeling
1. Introduction Ti/porcelain interfaces are usually found in materials for dental applications. Ti shows a good corrosion resistance as well as an excellent biocompatibility. The main problem associated with
*
Corresponding author. Address: Department of Chemistry and Materials, Universidad Europea de Madrid, (C) Tajo s/n, 28670 Villaviciosa de Odon, Madrid, Spain. Tel.: +34-912115658; fax: +34-91-6168265/6468565. E-mail addresses: [email protected] (I. LadoTouri~ no), [email protected] (F. Tsobnang). 1 Tel.: +33-24-3214001; fax: +33-24-3214039.
these interfaces is a decrease of bond strength between the porcelain and the Ti after a certain amount of wear [1]. Some thermodynamic calculations have shown that this phenomenon is caused by a high reactivity of Ti, particularly with O, which gives rise to the formation of scaly crystalline structures in the Ti-porcelain contact zone. These structures mainly consist of stoichiometric and non-stoichiometric titanium oxides [2] and are responsible of a continuing embrittlement of these interfaces [1,3–5]. It is therefore necessary to find a way to block the excessive reactivity of Ti with O. To achieve this, the implantation of Ti with Si ions seems to
0927-0256/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.05.005
14
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
be a suitable process [5,6] as the titanium silicide structures formed after the implantation process show a very high thermodynamic stability [7,8]. This stability prevents the formation of the scaly structures observed in conventional Ti/porcelain systems, which in turn gives rise to an increased bond strength between the Ti and the porcelain. Due to their complexity (existence of several phases, different kind of bonds, defects, dislocations, . . .) these interfaces have been investigated insufficiently or not at all on the atomic scale. By using molecular modeling techniques we want to get a better understanding and improve the physical and chemical properties of these materials. As a first step, in the present work we describe some calculations on the energetics of defects in Ti. Specifically, we have focused on the study of the energetics and electronic structure of an isolated O interstitial in a Ti crystal by ab initio methods. In the abundant literature on numerical calculations involving Ti–O bonding in both the bulk and especially the surface [9–14] we have found only small pieces of information regarding the energetics of O defects in Ti [15]. Nevertheless, these previous works demonstrate the capability of ab initio methods to study geometry relaxations, electronic structure and energetics (heats of formation, diffusion barriers, etc.) associated to point defects in crystals.
taken to complete convergence with respect to the size of the basis set. The main problem is the necessity of running calculations with prohibitively large basis sets in some cases. However, this problem can be avoided by using the so-called finite basis set correction [20], which corrects the results obtained at relatively low cutoff energy and k-point sampling. The only parameter that has to be known in order for this correction term to be evaluated can be expressed as dEtot =d lnEcut , where Etot is the total energy of the system and Ecut is the cutoff energy. Thus, this parameter gives a good indication of the convergence of the calculation with respect to the energy cutoff and k-point sampling. The electronic relaxation is performed by minimizing the sum of electronic eigenvalues in a fixed potential [21]. Brillouin-zone sampling is performed using the Monkhorst–Pack scheme [22]. Geometry optimization of all atomic positions is accomplished by the BFGS method, iterated until the root mean square force on each atom is less Calculations are performed in than 0.05 eV/A. periodic boundary conditions. As the main objective of this work is focused on the energetics of the isolated defects, each repeated cell contains a single O interstitial and 16 Ti atoms.
3. Results and discussion 3.1. Stability of interstitial sites
2. Method The present work was performed within the framework of density functional theory in the local density approximation for electronic exchange and correlation, the exchange and correlation energy as a function of electron density being that of Perdew and Zunger [16], as implemented in the CASTEP code [17]. The nuclei and core electrons are represented by the ultrasoft pseudopotentials generated using the optimization scheme of Vanderbilt [18]. All pseudopotentials are used in a separable Kleinman–Bylander form [19]. The valence electron pseudowave functions are expanded in terms of plane-waves whose kinetic energy is less than a chosen cutoff energy. The use of plane waves allows the calculation to be systematically
There are two possible interstitial sites for O in Ti: the octahedral site and the tetrahedral site (Fig. 1). It is known that oxygen is soluble in a-Ti with the O atoms being randomly distributed in the octahedral interstices of the hexagonally closepacked Ti lattice [23]. We have calculated the ground-state energies for O at these two sites, both with and without geometry optimization. The results for a cutoff energy of 600 eV for the non-relaxed case are shown in Table 1. Even if anything is gained from these unrelaxed models, we present them for the sake of comparison with the relaxed results. We also present in this table the embedding energy of the O atom in Ti, that is to say, the change of energy on going from perfect bulk Ti plus an
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
15
Fig. 1. O/Ti models used in the calculations. (A) Octahedral position and (B) tetrahedral position.
Table 1 Total ground-state energies and embedding energies for O in a 16-atom Ti crystal System
E (eV)
dEtot =d lnEcut (eV/atom)
Ee (eV) ðETiþO –ETi –EO Þ
Ti 16+oct. O Ti 16+tet. O
)26084.02 )26081.82
0.015 0.015
)11.09 )8.89
isolated O atom in its triplet ground-state to the Ti crystal with the O at an interstitial site. For the sake of comparison, the embedding energy calculated with two other all electron codes is
presented in Table 2 [24,25]. As one can see the absolute values of the embedding energy largely differ from one another but the three calculations show that the octahedral position is more stable in
16
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
accordance with the experimental results. The dEtot =d lnEcut values should be less than 0.01 eV/ atom for a calculation to be considered well converged. When the crystal is allowed to relax, nothing seems to happen for the octahedral case but it is worth noting the movement of the O atom after relaxing the structure in the tetrahedral case (Fig. 2). The values obtained for the relaxed cases are shown in Table 3. The tetrahedral site seems to be unstable. The oxygen atom moves down on to the plane of the Ti atoms becoming three-fold coordinated (Fig. 2). The Ti–O distance is 1.845 A (see Table 4). From this table one can note that in relaxing the atomic positions the Ti–O distances increase in the tetrahedral case and decrease a little in the octahedral case approaching the distances found in the most important titanium oxides (from in TiO2 crystals, which present 1.930 up to 1.982 A for a three-fold coordinated O atom, and 2 A Table 2 Embedding energies in eV for O in a 16-atom Ti crystal calculated with two different all-electrons codes System
Fast structure [17]
DMol [18]
Ti 16+oct. O Ti 16+tet. O
)16.05 )13.06
)10.07 )7.62
Table 3 Total ground-state energies and embedding energies for O in a 16-atom Ti crystal after relaxation of atomic positions System
E (eV)
Ti 16+oct. O )26084.66 Ti 16+tet. O )26083.02
dEtot =d lnEcut (eV/atom)
Ee (eV) ðETiþO –ETi –EO Þ
0.015 0.015
)11.73 )10.09
Table 4 Ti–O distances in angstroms in O containing Ti crystals Non-relaxed octahedral
Non-relaxed tetrahedral
Relaxed octahedral
Relaxed tetrahedral
2.066 (6)
1.803 (3) 1.749 (1)
2.045 (6)
1.845 (3) 2.169 (1)
The figures in the parentheses indicate the number of equivalent Ti–O bonds existing in the structure.
Ti2 O3 , which presents a four-fold coordinated O atom). 3.2. Electronic properties As a first step towards a complete understanding of the bonding characteristics at Ti/porcelain interfaces, the valence electron density and the
Fig. 2. Final position of an O atom initially in a tetrahedral site after relaxation of the atomic positions.
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
Mulliken charges for the O containing systems have been calculated. The electron density for different crystal planes containing O–Ti bonds is shown in Fig. 3. For the sake of comparison the results obtained for one of the polymorphs of TiO2 (rutile) are also displayed. The Mulliken charges are presented in Table 5. From these results one can see that there is a charge transfer from the Ti atoms to the O atom, which is more important for the nearest Ti atoms. In all the cases the Mulliken charge on the O atom is )1 electron. The nature
17
of the Ti–O bond is very similar to that found in titanium oxide crystals.
4. Conclusions and perspectives Our first calculations on O containing Ti crystals show that an O atom is more stable in the octahedral configuration than in the tetrahedral one. The same result is found using different types of codes, always within the framework of the
Fig. 3. Contour plots of the valence electron density on crystal planes containing Ti–O bonds. The values marked on the upper right indicate the electron number density in units of electron/bohr3 . (A) Rutile crystal, (B) octahedral position, (C) tetrahedral position, and (D) tetrahedral position after relaxation.
18
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
Fig. 3 (continued) Table 5 Mulliken charges on the O atom and its nearest neighbors Ti atoms System
qO
qTi
Non-relaxed tetrahedral
)1.066
Non-relaxed octahedral Relaxed tetrahedral
)0.941 )0.946
Relaxed octahedral
)0.943
0.200 0.333 0.143 0.150 0.132 0.145
(1.803), (1.749) (1.845), (2.169)
local-density approximation. Furthermore, the tetrahedral position seems to be unstable and after
relaxing the atomic positions the O atom becomes three-fold coordinated like in most TiO2 crystals. This may be due to the small distance between the O atom and its nearest neighbors in the tetrahedral position. The bonding characteristics are also similar to those existing in TiO2 crystals. It has to be noted that this is a preliminary study and the results are to be considered from a qualitative point of view only. The main sources of inaccuracy in the calculation are the plane-wave cutoff, the size of the cell and the Brillouin-zone sampling. Calculations with a larger cutoff energy and a more complete Brillouin-zone sampling may be necessary. The size of the repeating cell is also
I. Lado-Touri~no, F. Tsobnang / Computational Materials Science 32 (2005) 13–19
an important question that has to be considered [15]. The distance between the O atoms in the z for all the systems while in direction is about 9 A the x and y directions this distance decreases to This may give rise to significant electronic 5 A. interactions between the O atoms. Moreover, Brillouin-zone sampling errors and mainly lattice relaxation will be affected by the size of the system. Thus, calculations on larger cells are mandatory. The question of the origin of the oxygen is also important if one wants to obtain quantitative results. If the O atom comes from molecular oxygen, one could add to the embedding energy, the dissociation energy per atom of the oxygen molecule in order to obtain the heat of solution. This energy was calculated recently for an O interstitial in the octahedral position in a 96-atom supercell [15]. The value of )6.12 eV obtained from this calculation is not very different from the value that we obtain ()6.61 eV) when subtracting the value of the experimental dissociation energy (5.12 eV) from the value calculated for the embedding energy (see Table 3). More calculations have to be performed to be able to compare results from different research groups and set up quantitative conclusions. If one considers the O atom to be donated by the oxidic porcelain, more complicated redox reactions and charge states must be considered. In conclusion, this calculation is only a first step towards a complete understanding of the phenomena that take place at porcelain/Ti interfaces. Larger models are necessary and a more exhaustive analysis of the parameters used in the calculation has to be done if one wants to get realistic results.
19
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