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Yttrium segregation and oxygen diffusion along high-symmetry grain boundaries in YSZ
Accepted Manuscript Yttrium segregation and oxygen diffusion along high-symmetry grain boundaries in YSZ Robert L. González-Romero, Juan J. Meléndez PII: DOI: Reference:
S0925-8388(14)02622-X http://dx.doi.org/10.1016/j.jallcom.2014.10.184 JALCOM 32530
To appear in:
Journal of Alloys and Compounds
Received Date: Revised Date: Accepted Date:
2 September 2014 28 October 2014 29 October 2014
Please cite this article as: R.L. González-Romero, J.J. Meléndez, Yttrium segregation and oxygen diffusion along high-symmetry grain boundaries in YSZ, Journal of Alloys and Compounds (2014), doi: http://dx.doi.org/10.1016/ j.jallcom.2014.10.184
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YTTRIUM SEGREGATION AND OXYGEN DIFFUSION ALONG HIGH-SYMMETRY GRAIN BOUNDARIES IN YSZ Robert L. González-Romero1, Juan J. Meléndez2, 3 1
Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, Caixa Postal 6165, CEP 13083-970, Campinas (Brazil) 2
3
Department of Physics, University of Extremadura.
Institute for Advanced Scientific Computing of Extremadura (ICCAEx). Avda. de Elvas, s/n, 06006, Badajoz (Spain)
Abstract A study by Molecular Dynamics of yttrium segregation to high-symmetry grain boundaries of yttria-stabilized zirconia has been performed for different amounts of dopants. After an initial (and short) transient, segregation reaches a steady regime in which the concentration of the defect species at the grain-boundaries does not change in time. The maximum concentration of yttrium is reached at the grain-boundary planes, while oxygen vacancies screen the electric field created by segregation. Segregation of yttrium does not change appreciably the coefficients for oxygen diffusion along the grain boundaries, but instead modifies those for bulk diffusion. This effect is rationalized in terms of the rearrangement of the oxygen vacancies at the vicinities of the yttrium cations. The activation energies vary smoothly with the concentration of yttria for all the boundaries. Our data for diffusion coefficients and activation energies compare fairly well with experimental values when segregation is explicitly taken into account. Keywords: GB diffusion; yttrium segregation; Molecular Dynamics;YSZ 1. INTRODUCTION The role played by segregation of impurities in the yttria-zirconia system has long been recognized. Yttrium cations, which are aliovalent in the ZrO2 lattice, segregate to the grain boundaries (GBs) in yttria-zirconia ceramics by a mechanism which combines the elastic misfit effect (since the dissimilarity between the ionic radii of the Zr4+ (0.92 Å) and Y3+
1
(1.16 Å) cations helps the latter to segregate [1]) and the Coulombic interactions between the dopants and the oxygen vacancies which must be created for charge compensation [2]. Yttrium segregation is partially responsible for a number of exceptional mechanical and (especially) electrical properties exhibited by the yttria-zirconia system [3-6], which justifies the interest to investigate this phenomenon. The scientific literature contains several experimental studies at this respect [1, 7-13]; numerical simulation, which allows a much more accurate control of the geometric variables at the GB scale, has been used as well [14-17]. These works have put forth some evidences. Thus, it has been demonstrated that Y3+ segregate within a layer a few nanometers thick at the sides of a GB plane and that oxygen vacancies tend to co-segregate together with Y3+. Some other issues, such as the local atomic structure of segregated GBs or the effect of segregation on oxygen diffusion along the grain boundaries, remain controversial or unknown. In a previous paper, a preliminary Molecular Dynamics (MD) study about segregation of yttrium to a Σ5 (310)/[001] tilt grain boundary in yttria-stabilized cubic zirconia (YSZ) [18] was performed. It was shown there that the co-segregation of oxygen vacancies with yttrium cations was electrically and energetically favorable. In addition, it was pointed out that segregation did not have any appreciable influence on oxygen diffusion along the GBs, but could affect oxygen bulk diffusion. This fact, which has obvious effects in ionic conduction, is more closely examined in this work, where we extend the previous one to several other GBs with different geometries and for various concentrations of dopants for the sake of generality. In particular, it will be shown that segregation does not change appreciably the characteristics of GB diffusion of oxygen in this system. This fact will be correlated to the distribution of segregated yttrium-oxygen vacancy pairs at the GBs, as well as the preferential locations of these vacancies. As a consequence, it will be demonstrated that segregation affects bulk diffusion; the validity of this conclusion will be addressed by comparison with experimental data.
2
2. METHODOLOGY Three models for YSZ bicrystals, each one containing two symmetric grain boundaries (namely Σ11, Σ13 and Σ3) were built; details about each boundary geometry can be found elsewhere [19]. Two regions were distinguished in each bicrystal. The ions within 6 Å at each side of each GB were defined as “grain boundary ions”. The data concerning GBs reported herein (oxygen diffusion coefficients and activation energies, coordination numbers) were calculated considering only these ions. All the others were considered as “bulk ions”, from which bulk properties were calculated. Yttria concentrations were set to 8, 12, 15 and 18 mol %, with yttrium cations replacing zirconium ones; the corresponding bicrystals will be labeled as 8Y, 12Y, 15Y and 18Y, respectively. For each simulation box, the proper numbers of oxygen vacancies were subsequently introduced to keep the electrical neutrality of the system. The dopants and vacancies substitutions were randomly made. For each composition and GB, five different configurations were considered for statistical purposes; the reported data were taken as the average over this population of five samples. MD runs were performed by the LAMMPS code [20]. The potential of interaction between
ions and was modeled as a Buckingham-type one coupled with a long-range Coulombic term: () = –
− +
(1)
where is the interionic distance and ! holds for the charge of ion . This potential model has been successfully employed to describe the dynamical behavior of several
systems, including grain boundaries in YSZ [19]. The parameters , " and # (cf. table I) were taken from the literature [21, 22]; the charges of each ion were taken as the full ones. Each initial configuration was optimized by energy minimization followed by a 10 ps run of MD at constant volume with $% = 0 Pa and '% = 0 K. After minimization, the temperature was set to 1500 K and MD calculations were subsequently carried out for times up to 30 ps to induce segregation; a set of simulations was carried out at higher 3
temperatures, but no differences with respect to the lowest one were found. To simulate diffusion of oxygen, temperatures within the range 1500 K – 3000 K were set after segregation, and runs were performed during 200 ps. In all cases, the NVT ensemble with $ = 0 Pa was used.
The interaction potential [Eq. (1)] was originally developed to study solution mechanisms and defect cluster geometries for several dopant cations into a ceramic matrix. Therefore, it was not expected to yield segregation by itself despite the use of long simulation times (up to 500 ps [19]). Previous theoretical and simulation studies have shown that segregation is driven, at least at the first stages, by the elastic misfit effect of the yttrium cations located substitutionally within the cationic sublattice in YSZ [17, 23, 24]. As in the previous work [18], and following Yan and co-workers [23], yttrium cations within a bin of thickness ) in the *+, plane were accepted to have an excess segregation energy given by: ∆1
∆.(/) = 0 ∆12 2
(/ − /3 + )) for /3 − ) < / ≤ /3
(/3 + ) − /) for /3 ≤ / < /3 + )
9
(2)
where ∆.% is the excess segregation energy at the grain boundary and /3 is the position of
the grain boundary along the OZ axis. In YSZ, ) ≈ 0.1=> typically [2]; for this work, it was taken as ) ≈ 6 Å. On the other hand, the segregation energy at the GBs has been estimated as ∆.% ≈ 0.5 eV at temperatures above 1473 K [23, 24]. These choices for ∆.% and )
sufficed for yttrium cations to segregate at the GBs, and also allowed us to follow the evolution of the different defect species within the simulation box, as was reported elsewhere [18]. Anyway, we notice here that the results reported below are not much
dependent on the particular choices of either ∆.% (which affects essentially to the kinetics, that is, to the time required for the yttrium cations to segregate to the GBS) or ) (which affects to the redistribution of ionic species following segregation). The radial distribution functions (RDFs) and coordination numbers for ion pairs were computed referenced to yttrium cations in the GB and bulk regions. For diffusion of
oxygen, the mean square displacements 〈 B 〉 of oxygen anions within the GB and bulk 4
regions were computed throughout each run as functions of the simulation time D. The corresponding diffusion coefficients were calculated from these displacements as [25]: E = limI→K
〈L 〉 MI
(3)
3. RESULTS In all the GBs and for all the concentrations, a steady regime, which we identify as equilibrium segregation, was reached after a relatively short transient period, typically around 10-30 ps in the conditions of this work. During the transient regime, the energy by ion computed at the grain boundaries decreased monotonically, which indicates that segregation is energetically favorable. In what follows, only data obtained within the steady regime will be reported.
× U Figs. 1 plot the numbers of each type of defect (ZrPQ , YPQ and vW·· ) relative to the nominal
ones vs. the reduced distance (that is, the distance along the OZ axis from the origin of the simulation box over Lz) calculated at 1500 K in an arbitrary configuration of 8Y for Σ11 (a), Σ13 (b) and Σ3 (c); dashed vertical lines denote the positions of the grain boundaries. From these plots, it is evident that the maximum concentration of yttrium is reached at the grain boundary plane, at the expenses of that in the adjacent atomic layers. In addition, the concentrations of zirconium and oxygen vacancies have minima at the GB planes and maxima at short distances at each side of them. Therefore, figs. 1 indicate that adding the excess energy [Eq. (2)] is effective in simulating segregation. Indeed, segregation of aliovalent yttrium implies an excess of negative charge at the vicinity of the grain boundaries, which must be screened somehow as one moves towards the grain interior. In this system, only oxygen vacancies are effective to screen the GB electrical potential, so that they must locate within the excess of charge at the GB and the grain interior; this is precisely the effect shown in Figs. 1, which is in good agreement with experimental evidence [11-13]. The distribution of zirconium, on the other hand, results from the fact that yttrium incorporates substitutionally to the lattice of ZrO2; Figs. 1 suggest then that the zirconium cations expelled from the GB planes by the segregating yttrium tend to 5
locate within the few atomic layers at each side of the boundaries. It is noteworthy that the excess energy is added only to yttrium cations; thus, the coupled movement of zirconium and oxygen vacancies arises naturally from their mutual electrostatic interaction. Fig. 2 plots the average relative number of yttrium cations at the GB planes vs. the concentration of yttria for the three boundaries. These numbers range roughly between 2
and 3.5, which correspond to enrichment factors (defined as the ratios
YZ][\ ^ h _`abcdefg ijkljkmnop ] 9 qZ[\ r h _`abcdefg so tjkljkmnop
9
,
where the square brackets denote concentrations) between 2.6 and 3.4. The relative number of substitutional cations increases with the concentration of dopant for Σ11 and Σ3, whereas it decreases for Σ13. This result suggests that some of the segregated cations could locate interstitially in the last case. In addition, the relative number of yttrium at the GBs does not vary with temperature, as exemplified in Fig. 3 for Σ3. Both facts agree reasonably well, given the simplicity of our model, with simulations and experiments reported elsewhere [1, 7-10, 15, 17, 26-34]. Fig. 4 plots the coefficients of oxygen diffusion along the Σ11 GB vs. the reciprocal temperature calculated after segregation for different concentrations of dopants; open symbols correspond to data calculated within this work, and the solid lines with the same color code are those of best fit to the data for GB oxygen diffusion without segregation, as reported elsewhere [19]. This plot demonstrates that segregation of yttrium does not change appreciably the coefficients of oxygen diffusion along the Σ11 grain boundary; this effect was found in all the other GBs as well (although data are not shown here for brevity), and suggests that the trend about Σ5 published elsewhere [18] may be actually of general validity. Fig. 5 plots the activation energy for GB diffusion of oxygen as function of the concentration of dopant calculated for each boundary after segregation; data for unsegregated boundaries are included as solid lines for comparison. These activation energies no longer exhibit a maximum for 15 mol % [19], but vary smoothly with the concentration of yttria. 6
4. DISCUSSION Oxygen diffusion in bulk YSZ has been very widely studied since, as mentioned in the Introduction, it is a crucial subject to understand some of the physical properties of this system. At this respect, numerical simulation (in particular Molecular Dynamics) has shown that oxygen diffusion is intimately related to the distribution of oxygen vacancies within the YSZ lattice [19, 35, 36]. Since one of the most striking findings of this work is the fact that grain boundary diffusion of oxygen is practically unaffected by segregation, we have performed an analysis of the distribution of chemical species at the vicinities of the GBs as well as in bulk. The observed trends about diffusion coefficients may be understood as consequence of the rearrangement of oxygen vacancies after segregation. Indeed, it has been reported that
U oxygen vacancies get trapped when they locate as second neighbors of isolated YPQ defects U U in bulk YSZ [35]; on the other hand, YPQ − YPQ clusters act as effective traps for oxygen
U vacancies if these locate as first neighbors of the YPQ cations forming neutral multiple U U g× defects of the form _YPQ − vu·· − YPQ [36]; the relative location of oxygen vacancies with
× respect to ZrPQ , on the other hand, seems to have no effect on diffusion. Our simulations
show that segregation increases the number of yttrium cations which locate as first neighbors to each other. This effect is demonstrated in table 2, where the average coordination numbers at the first and second coordination spheres are shown for pairs U U YPQ − YPQ before and after segregation. Surprisingly, these numbers do not depend on the
U particular boundary geometry, but only on the composition. Regarding YPQ − O× W pairs, the
coordination numbers at the first coordination sphere decrease upon segregation for all concentrations, whereas they increase at the second coordination sphere. To illustrate this effect, Fig. 6 shows the coordination numbers at the first and second coordination spheres as functions of the concentration of dopant calculated in Σ11; the same trend appears in all the other boundaries.
7
U In other words, after segregation each YPQ at the GBs is surrounded in average by more
oxygen vacancies as first neighbors, which are little mobile. Most of these vacancies were U located before segregation as second neighbors of the YPQ defects, where they were also
little mobile, as shown in Fig. 6. Thus, segregation just causes the rearrangement of little U mobile oxygen vacancies from second neighbors of isolated YPQ before segregation to first
U U neighbors of the highly probable YPQ − YPQ clusters after segregation, which justifies why
the coefficients of oxygen diffusion do not change much with segregation. But some other oxygen vacancies must come to the GBs from the grain interior as segregation takes place. We have already mentioned that the electric field induced by the extra segregated yttrium cations must be screened by mobile defects with positive charge; in our system, these are mainly oxygen vacancies, which must be mobile. These vacancies are likely to yield the concentration maxima shown in Figs. 1. This effect is quantitatively U demonstrated in Fig. 7, which plots the coordination numbers for YPQ − O× W pairs at the
first and second coordination spheres, calculated for bulk ions, as functions of the concentration of dopant. In this figure one observes that migration of oxygen vacancies U from the grain interior to the boundaries affects mainly the first neighbors of YPQ − O× W
pairs. For the second neighbors, on the contrary, the variation is much noticeably smaller; note that most of the vacancies at these positions are immobile [35]. The previous paragraph suggests that segregation to the GBs must affect oxygen bulk diffusion. To verify this hypothesis, the oxygen bulk diffusion coefficients have been calculated before and after segregation, and the results summarized in the Arrhenius plots Fig. 8; open symbols correspond to averaged data calculated in the bulk regions of the three bicrystals after segregation, and solid lines are those of best fit for data calculated before segregation. The analysis of the results is easier if the differences between the bulk diffusion coefficients before and after segregation are plotted as function of the temperature, as in fig. 9. From it one sees that, for concentrations below 18 mol %, the difference between diffusion coefficients increases with temperature to reach a maximum 8
at some critical temperature which depends on the concentration of dopant. At temperatures above the critical, the difference between coefficients decreases monotonically. For 18 mol %, the oxygen bulk diffusion coefficients for segregated boundaries are lower than those before segregation at all temperatures. Segregation affects the activation energies for bulk diffusion as well. Table 3 records these energies, calculated in the bulk region before and after segregation. Segregation increases the activation energies, the difference with those in absence of segregation decreasing with the concentration of dopant. The issue remains as to compare our calculations with experimental data. The activation energies for oxygen bulk diffusion measured either by oxygen tracer diffusion [37-39] or by impedance spectroscopy [37, 40] range between 0.82 eV and 1.10 eV for 8 mol % or 9.5 mol % Y2O3. These values are systematically higher than those obtained from MD or kinetic Monte Carlo simulations, which range instead between 0.45 eV and 0.8 eV (see [19] and references therein), but compare fairly well with those reported in this work after segregation has taken place. This fact was expected, since segregation appears as consequence of the fabrication processes in real samples. Moreover, the difference between the activation energies for GB and bulk diffusion, wx3 − wyz2{ , measured by different experimental techniques, lies in the range 0.05 – 0.36 eV (see, for instance, [41] and references therein). Fig. 10 plots our calculated values for wx3 − wyz2{ before and after segregation as functions of the concentration of dopants; the horizontal lines correspond to the experimental limits. From fig. 10 it is clear that, when segregation is explicitly taken into account in the computational model, numerical and experimental results agree reasonably well, whereas data for unsegregated boundaries lie mostly outside the experimental range. ACKNOWLEDGEMENTS This work has been financially supported by the Ministerio de Economía y Competitividad, Government of Spain, through Grant MAT2012-38205-C02-02. The authors acknowledge 9
Dr. Carlos J. García Orellana, from the University of Extremadura, for his helpful computational support. REFERENCES
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Figure captions Figs. 1: Number of defects (relative to the nominal one) vs. reduced distance for the Σ11 (a), Σ13 (b) and Σ3 (c) GBs (arbitrary configurations) calculated at 1500 K in 8Y. Fig. 2: Average relative number of yttrium cations at the GB planes vs. the concentration of yttria for the three boundaries. Fig. 3: Average relative number of yttrium cations at the GB planes vs. temperature for the Σ3 GB (as reference) for different concentrations of yttria. Fig. 4: Arrhenius plot for the coefficients of oxygen diffusion along Σ11 for different concentrations of dopant. Results for Σ11 without segregation are included as straight lines for comparison. Fig. 5: Activation energy for oxygen diffusion along the GBs vs. concentration of yttria. Data calculated in absence of segregation are also included for comparison. U Fig. 6: Coordination numbers for YPQ − O× W pairs at the first or second coordination
spheres, as functions of the concentration of dopant, calculated in Σ11 before and after segregation. U Fig. 7: Coordination numbers for YPQ − O× W pairs at the first or second coordination
spheres, as functions of the concentration of dopant, calculated in in the bulk region before and after segregation. Fig. 8: Arrhenius plot for the coefficients of oxygen bulk diffusion calculated for different concentrations of dopant before (as solid lines, see text) and after (individual symbols) segregation. Fig. 9: Difference between bulk diffusion coefficients before and after segregation as functions of the temperature for several concentrations of dopant.
13
Fig. 10: Difference between the activation energies for GB- or bulk diffusion of oxygen vs. concentration of yttria calculated after (filled symbols) or in absence of segregation (open ones). Values for wx3 were taken as an average over the three GBs.
14
Table 1: Parameters for potential [Eq. (1)] used in this study (from [18, 19]) Ion pair
³´µ (eV)
¶´µ (Å)
·´µ (eV·Å6)
O2- - O2-
9547.96
0.2192
32.0
Zr4+ - O2-
1502.11
0.3477
5.1
Y3+ - O2-
1766.40
0.33849
19.43
15
Table 2: Average number of the coordination numbers in the first and second U U coordination spheres for pairs YPQ − YPQ before and after segregation.
Before segregation
After segregation
1st sphere
2nd sphere
1st sphere
2 nd sphere
8Y
1.43 ± 0.09
0.71 ± 0.10
2.61 ± 0.11
1.02 ± 0.12
12Y
2.05 ± 0.11
1.12 ± 0.09
2.87 ± 0.13
1.39 ± 0.11
15Y
2.54 ± 0.09
1.25 ± 0.10
3.45 ± 0.11
1.54 ± 0.10
18Y
2.96 ± 0.12
1.60 ± 0.08
3.72 ± 0.10
1.82 ± 0.13
16
Table 3: Activation energies (in eV) for bulk diffusion calculated before and after segregation. Before segregation
After segregation
8Y
0.71 ± 0.04
0.88 ± 0.03
12Y
0.80 ± 0.02
0.95 ± 0.03
15Y
0.88 ± 0.03
1.01 ± 0.04
18Y
1.03 ± 0.03
1.04 ± 0.02
17
RESEARCH HIGHLIGHTS FOR
YTTRIUM SEGREGATION AND OXYGEN DIFFUSION ALONG
HIGH-SYMMETRY GRAIN BOUNDARIES IN YSZ
by Robert L. González-Romero and Juan J. Meléndez
-
A study of yttrium segregation to three grain boundaries (GB) in YSZ is performed.
-
A steady concentration of yttrium is reached after a short transient regime.
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Segregation does not change the coefficients for oxygen diffusion along the GB.
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The main effect after segregation appears for oxygen bulk diffusion.
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The effect is related to the rearrangement of oxygen vacancies after segregation.
18
S0925-8388(14)02622-X http://dx.doi.org/10.1016/j.jallcom.2014.10.184 JALCOM 32530
To appear in:
Journal of Alloys and Compounds
Received Date: Revised Date: Accepted Date:
2 September 2014 28 October 2014 29 October 2014
Please cite this article as: R.L. González-Romero, J.J. Meléndez, Yttrium segregation and oxygen diffusion along high-symmetry grain boundaries in YSZ, Journal of Alloys and Compounds (2014), doi: http://dx.doi.org/10.1016/ j.jallcom.2014.10.184
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YTTRIUM SEGREGATION AND OXYGEN DIFFUSION ALONG HIGH-SYMMETRY GRAIN BOUNDARIES IN YSZ Robert L. González-Romero1, Juan J. Meléndez2, 3 1
Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, Caixa Postal 6165, CEP 13083-970, Campinas (Brazil) 2
3
Department of Physics, University of Extremadura.
Institute for Advanced Scientific Computing of Extremadura (ICCAEx). Avda. de Elvas, s/n, 06006, Badajoz (Spain)
Abstract A study by Molecular Dynamics of yttrium segregation to high-symmetry grain boundaries of yttria-stabilized zirconia has been performed for different amounts of dopants. After an initial (and short) transient, segregation reaches a steady regime in which the concentration of the defect species at the grain-boundaries does not change in time. The maximum concentration of yttrium is reached at the grain-boundary planes, while oxygen vacancies screen the electric field created by segregation. Segregation of yttrium does not change appreciably the coefficients for oxygen diffusion along the grain boundaries, but instead modifies those for bulk diffusion. This effect is rationalized in terms of the rearrangement of the oxygen vacancies at the vicinities of the yttrium cations. The activation energies vary smoothly with the concentration of yttria for all the boundaries. Our data for diffusion coefficients and activation energies compare fairly well with experimental values when segregation is explicitly taken into account. Keywords: GB diffusion; yttrium segregation; Molecular Dynamics;YSZ 1. INTRODUCTION The role played by segregation of impurities in the yttria-zirconia system has long been recognized. Yttrium cations, which are aliovalent in the ZrO2 lattice, segregate to the grain boundaries (GBs) in yttria-zirconia ceramics by a mechanism which combines the elastic misfit effect (since the dissimilarity between the ionic radii of the Zr4+ (0.92 Å) and Y3+
1
(1.16 Å) cations helps the latter to segregate [1]) and the Coulombic interactions between the dopants and the oxygen vacancies which must be created for charge compensation [2]. Yttrium segregation is partially responsible for a number of exceptional mechanical and (especially) electrical properties exhibited by the yttria-zirconia system [3-6], which justifies the interest to investigate this phenomenon. The scientific literature contains several experimental studies at this respect [1, 7-13]; numerical simulation, which allows a much more accurate control of the geometric variables at the GB scale, has been used as well [14-17]. These works have put forth some evidences. Thus, it has been demonstrated that Y3+ segregate within a layer a few nanometers thick at the sides of a GB plane and that oxygen vacancies tend to co-segregate together with Y3+. Some other issues, such as the local atomic structure of segregated GBs or the effect of segregation on oxygen diffusion along the grain boundaries, remain controversial or unknown. In a previous paper, a preliminary Molecular Dynamics (MD) study about segregation of yttrium to a Σ5 (310)/[001] tilt grain boundary in yttria-stabilized cubic zirconia (YSZ) [18] was performed. It was shown there that the co-segregation of oxygen vacancies with yttrium cations was electrically and energetically favorable. In addition, it was pointed out that segregation did not have any appreciable influence on oxygen diffusion along the GBs, but could affect oxygen bulk diffusion. This fact, which has obvious effects in ionic conduction, is more closely examined in this work, where we extend the previous one to several other GBs with different geometries and for various concentrations of dopants for the sake of generality. In particular, it will be shown that segregation does not change appreciably the characteristics of GB diffusion of oxygen in this system. This fact will be correlated to the distribution of segregated yttrium-oxygen vacancy pairs at the GBs, as well as the preferential locations of these vacancies. As a consequence, it will be demonstrated that segregation affects bulk diffusion; the validity of this conclusion will be addressed by comparison with experimental data.
2
2. METHODOLOGY Three models for YSZ bicrystals, each one containing two symmetric grain boundaries (namely Σ11, Σ13 and Σ3) were built; details about each boundary geometry can be found elsewhere [19]. Two regions were distinguished in each bicrystal. The ions within 6 Å at each side of each GB were defined as “grain boundary ions”. The data concerning GBs reported herein (oxygen diffusion coefficients and activation energies, coordination numbers) were calculated considering only these ions. All the others were considered as “bulk ions”, from which bulk properties were calculated. Yttria concentrations were set to 8, 12, 15 and 18 mol %, with yttrium cations replacing zirconium ones; the corresponding bicrystals will be labeled as 8Y, 12Y, 15Y and 18Y, respectively. For each simulation box, the proper numbers of oxygen vacancies were subsequently introduced to keep the electrical neutrality of the system. The dopants and vacancies substitutions were randomly made. For each composition and GB, five different configurations were considered for statistical purposes; the reported data were taken as the average over this population of five samples. MD runs were performed by the LAMMPS code [20]. The potential of interaction between
ions and was modeled as a Buckingham-type one coupled with a long-range Coulombic term: () = –
− +
(1)
where is the interionic distance and ! holds for the charge of ion . This potential model has been successfully employed to describe the dynamical behavior of several
systems, including grain boundaries in YSZ [19]. The parameters , " and # (cf. table I) were taken from the literature [21, 22]; the charges of each ion were taken as the full ones. Each initial configuration was optimized by energy minimization followed by a 10 ps run of MD at constant volume with $% = 0 Pa and '% = 0 K. After minimization, the temperature was set to 1500 K and MD calculations were subsequently carried out for times up to 30 ps to induce segregation; a set of simulations was carried out at higher 3
temperatures, but no differences with respect to the lowest one were found. To simulate diffusion of oxygen, temperatures within the range 1500 K – 3000 K were set after segregation, and runs were performed during 200 ps. In all cases, the NVT ensemble with $ = 0 Pa was used.
The interaction potential [Eq. (1)] was originally developed to study solution mechanisms and defect cluster geometries for several dopant cations into a ceramic matrix. Therefore, it was not expected to yield segregation by itself despite the use of long simulation times (up to 500 ps [19]). Previous theoretical and simulation studies have shown that segregation is driven, at least at the first stages, by the elastic misfit effect of the yttrium cations located substitutionally within the cationic sublattice in YSZ [17, 23, 24]. As in the previous work [18], and following Yan and co-workers [23], yttrium cations within a bin of thickness ) in the *+, plane were accepted to have an excess segregation energy given by: ∆1
∆.(/) = 0 ∆12 2
(/ − /3 + )) for /3 − ) < / ≤ /3
(/3 + ) − /) for /3 ≤ / < /3 + )
9
(2)
where ∆.% is the excess segregation energy at the grain boundary and /3 is the position of
the grain boundary along the OZ axis. In YSZ, ) ≈ 0.1=> typically [2]; for this work, it was taken as ) ≈ 6 Å. On the other hand, the segregation energy at the GBs has been estimated as ∆.% ≈ 0.5 eV at temperatures above 1473 K [23, 24]. These choices for ∆.% and )
sufficed for yttrium cations to segregate at the GBs, and also allowed us to follow the evolution of the different defect species within the simulation box, as was reported elsewhere [18]. Anyway, we notice here that the results reported below are not much
dependent on the particular choices of either ∆.% (which affects essentially to the kinetics, that is, to the time required for the yttrium cations to segregate to the GBS) or ) (which affects to the redistribution of ionic species following segregation). The radial distribution functions (RDFs) and coordination numbers for ion pairs were computed referenced to yttrium cations in the GB and bulk regions. For diffusion of
oxygen, the mean square displacements 〈 B 〉 of oxygen anions within the GB and bulk 4
regions were computed throughout each run as functions of the simulation time D. The corresponding diffusion coefficients were calculated from these displacements as [25]: E = limI→K
〈L 〉 MI
(3)
3. RESULTS In all the GBs and for all the concentrations, a steady regime, which we identify as equilibrium segregation, was reached after a relatively short transient period, typically around 10-30 ps in the conditions of this work. During the transient regime, the energy by ion computed at the grain boundaries decreased monotonically, which indicates that segregation is energetically favorable. In what follows, only data obtained within the steady regime will be reported.
× U Figs. 1 plot the numbers of each type of defect (ZrPQ , YPQ and vW·· ) relative to the nominal
ones vs. the reduced distance (that is, the distance along the OZ axis from the origin of the simulation box over Lz) calculated at 1500 K in an arbitrary configuration of 8Y for Σ11 (a), Σ13 (b) and Σ3 (c); dashed vertical lines denote the positions of the grain boundaries. From these plots, it is evident that the maximum concentration of yttrium is reached at the grain boundary plane, at the expenses of that in the adjacent atomic layers. In addition, the concentrations of zirconium and oxygen vacancies have minima at the GB planes and maxima at short distances at each side of them. Therefore, figs. 1 indicate that adding the excess energy [Eq. (2)] is effective in simulating segregation. Indeed, segregation of aliovalent yttrium implies an excess of negative charge at the vicinity of the grain boundaries, which must be screened somehow as one moves towards the grain interior. In this system, only oxygen vacancies are effective to screen the GB electrical potential, so that they must locate within the excess of charge at the GB and the grain interior; this is precisely the effect shown in Figs. 1, which is in good agreement with experimental evidence [11-13]. The distribution of zirconium, on the other hand, results from the fact that yttrium incorporates substitutionally to the lattice of ZrO2; Figs. 1 suggest then that the zirconium cations expelled from the GB planes by the segregating yttrium tend to 5
locate within the few atomic layers at each side of the boundaries. It is noteworthy that the excess energy is added only to yttrium cations; thus, the coupled movement of zirconium and oxygen vacancies arises naturally from their mutual electrostatic interaction. Fig. 2 plots the average relative number of yttrium cations at the GB planes vs. the concentration of yttria for the three boundaries. These numbers range roughly between 2
and 3.5, which correspond to enrichment factors (defined as the ratios
YZ][\ ^ h _`abcdefg ijkljkmnop ] 9 qZ[\ r h _`abcdefg so tjkljkmnop
9
,
where the square brackets denote concentrations) between 2.6 and 3.4. The relative number of substitutional cations increases with the concentration of dopant for Σ11 and Σ3, whereas it decreases for Σ13. This result suggests that some of the segregated cations could locate interstitially in the last case. In addition, the relative number of yttrium at the GBs does not vary with temperature, as exemplified in Fig. 3 for Σ3. Both facts agree reasonably well, given the simplicity of our model, with simulations and experiments reported elsewhere [1, 7-10, 15, 17, 26-34]. Fig. 4 plots the coefficients of oxygen diffusion along the Σ11 GB vs. the reciprocal temperature calculated after segregation for different concentrations of dopants; open symbols correspond to data calculated within this work, and the solid lines with the same color code are those of best fit to the data for GB oxygen diffusion without segregation, as reported elsewhere [19]. This plot demonstrates that segregation of yttrium does not change appreciably the coefficients of oxygen diffusion along the Σ11 grain boundary; this effect was found in all the other GBs as well (although data are not shown here for brevity), and suggests that the trend about Σ5 published elsewhere [18] may be actually of general validity. Fig. 5 plots the activation energy for GB diffusion of oxygen as function of the concentration of dopant calculated for each boundary after segregation; data for unsegregated boundaries are included as solid lines for comparison. These activation energies no longer exhibit a maximum for 15 mol % [19], but vary smoothly with the concentration of yttria. 6
4. DISCUSSION Oxygen diffusion in bulk YSZ has been very widely studied since, as mentioned in the Introduction, it is a crucial subject to understand some of the physical properties of this system. At this respect, numerical simulation (in particular Molecular Dynamics) has shown that oxygen diffusion is intimately related to the distribution of oxygen vacancies within the YSZ lattice [19, 35, 36]. Since one of the most striking findings of this work is the fact that grain boundary diffusion of oxygen is practically unaffected by segregation, we have performed an analysis of the distribution of chemical species at the vicinities of the GBs as well as in bulk. The observed trends about diffusion coefficients may be understood as consequence of the rearrangement of oxygen vacancies after segregation. Indeed, it has been reported that
U oxygen vacancies get trapped when they locate as second neighbors of isolated YPQ defects U U in bulk YSZ [35]; on the other hand, YPQ − YPQ clusters act as effective traps for oxygen
U vacancies if these locate as first neighbors of the YPQ cations forming neutral multiple U U g× defects of the form _YPQ − vu·· − YPQ [36]; the relative location of oxygen vacancies with
× respect to ZrPQ , on the other hand, seems to have no effect on diffusion. Our simulations
show that segregation increases the number of yttrium cations which locate as first neighbors to each other. This effect is demonstrated in table 2, where the average coordination numbers at the first and second coordination spheres are shown for pairs U U YPQ − YPQ before and after segregation. Surprisingly, these numbers do not depend on the
U particular boundary geometry, but only on the composition. Regarding YPQ − O× W pairs, the
coordination numbers at the first coordination sphere decrease upon segregation for all concentrations, whereas they increase at the second coordination sphere. To illustrate this effect, Fig. 6 shows the coordination numbers at the first and second coordination spheres as functions of the concentration of dopant calculated in Σ11; the same trend appears in all the other boundaries.
7
U In other words, after segregation each YPQ at the GBs is surrounded in average by more
oxygen vacancies as first neighbors, which are little mobile. Most of these vacancies were U located before segregation as second neighbors of the YPQ defects, where they were also
little mobile, as shown in Fig. 6. Thus, segregation just causes the rearrangement of little U mobile oxygen vacancies from second neighbors of isolated YPQ before segregation to first
U U neighbors of the highly probable YPQ − YPQ clusters after segregation, which justifies why
the coefficients of oxygen diffusion do not change much with segregation. But some other oxygen vacancies must come to the GBs from the grain interior as segregation takes place. We have already mentioned that the electric field induced by the extra segregated yttrium cations must be screened by mobile defects with positive charge; in our system, these are mainly oxygen vacancies, which must be mobile. These vacancies are likely to yield the concentration maxima shown in Figs. 1. This effect is quantitatively U demonstrated in Fig. 7, which plots the coordination numbers for YPQ − O× W pairs at the
first and second coordination spheres, calculated for bulk ions, as functions of the concentration of dopant. In this figure one observes that migration of oxygen vacancies U from the grain interior to the boundaries affects mainly the first neighbors of YPQ − O× W
pairs. For the second neighbors, on the contrary, the variation is much noticeably smaller; note that most of the vacancies at these positions are immobile [35]. The previous paragraph suggests that segregation to the GBs must affect oxygen bulk diffusion. To verify this hypothesis, the oxygen bulk diffusion coefficients have been calculated before and after segregation, and the results summarized in the Arrhenius plots Fig. 8; open symbols correspond to averaged data calculated in the bulk regions of the three bicrystals after segregation, and solid lines are those of best fit for data calculated before segregation. The analysis of the results is easier if the differences between the bulk diffusion coefficients before and after segregation are plotted as function of the temperature, as in fig. 9. From it one sees that, for concentrations below 18 mol %, the difference between diffusion coefficients increases with temperature to reach a maximum 8
at some critical temperature which depends on the concentration of dopant. At temperatures above the critical, the difference between coefficients decreases monotonically. For 18 mol %, the oxygen bulk diffusion coefficients for segregated boundaries are lower than those before segregation at all temperatures. Segregation affects the activation energies for bulk diffusion as well. Table 3 records these energies, calculated in the bulk region before and after segregation. Segregation increases the activation energies, the difference with those in absence of segregation decreasing with the concentration of dopant. The issue remains as to compare our calculations with experimental data. The activation energies for oxygen bulk diffusion measured either by oxygen tracer diffusion [37-39] or by impedance spectroscopy [37, 40] range between 0.82 eV and 1.10 eV for 8 mol % or 9.5 mol % Y2O3. These values are systematically higher than those obtained from MD or kinetic Monte Carlo simulations, which range instead between 0.45 eV and 0.8 eV (see [19] and references therein), but compare fairly well with those reported in this work after segregation has taken place. This fact was expected, since segregation appears as consequence of the fabrication processes in real samples. Moreover, the difference between the activation energies for GB and bulk diffusion, wx3 − wyz2{ , measured by different experimental techniques, lies in the range 0.05 – 0.36 eV (see, for instance, [41] and references therein). Fig. 10 plots our calculated values for wx3 − wyz2{ before and after segregation as functions of the concentration of dopants; the horizontal lines correspond to the experimental limits. From fig. 10 it is clear that, when segregation is explicitly taken into account in the computational model, numerical and experimental results agree reasonably well, whereas data for unsegregated boundaries lie mostly outside the experimental range. ACKNOWLEDGEMENTS This work has been financially supported by the Ministerio de Economía y Competitividad, Government of Spain, through Grant MAT2012-38205-C02-02. The authors acknowledge 9
Dr. Carlos J. García Orellana, from the University of Extremadura, for his helpful computational support. REFERENCES
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Figure captions Figs. 1: Number of defects (relative to the nominal one) vs. reduced distance for the Σ11 (a), Σ13 (b) and Σ3 (c) GBs (arbitrary configurations) calculated at 1500 K in 8Y. Fig. 2: Average relative number of yttrium cations at the GB planes vs. the concentration of yttria for the three boundaries. Fig. 3: Average relative number of yttrium cations at the GB planes vs. temperature for the Σ3 GB (as reference) for different concentrations of yttria. Fig. 4: Arrhenius plot for the coefficients of oxygen diffusion along Σ11 for different concentrations of dopant. Results for Σ11 without segregation are included as straight lines for comparison. Fig. 5: Activation energy for oxygen diffusion along the GBs vs. concentration of yttria. Data calculated in absence of segregation are also included for comparison. U Fig. 6: Coordination numbers for YPQ − O× W pairs at the first or second coordination
spheres, as functions of the concentration of dopant, calculated in Σ11 before and after segregation. U Fig. 7: Coordination numbers for YPQ − O× W pairs at the first or second coordination
spheres, as functions of the concentration of dopant, calculated in in the bulk region before and after segregation. Fig. 8: Arrhenius plot for the coefficients of oxygen bulk diffusion calculated for different concentrations of dopant before (as solid lines, see text) and after (individual symbols) segregation. Fig. 9: Difference between bulk diffusion coefficients before and after segregation as functions of the temperature for several concentrations of dopant.
13
Fig. 10: Difference between the activation energies for GB- or bulk diffusion of oxygen vs. concentration of yttria calculated after (filled symbols) or in absence of segregation (open ones). Values for wx3 were taken as an average over the three GBs.
14
Table 1: Parameters for potential [Eq. (1)] used in this study (from [18, 19]) Ion pair
³´µ (eV)
¶´µ (Å)
·´µ (eV·Å6)
O2- - O2-
9547.96
0.2192
32.0
Zr4+ - O2-
1502.11
0.3477
5.1
Y3+ - O2-
1766.40
0.33849
19.43
15
Table 2: Average number of the coordination numbers in the first and second U U coordination spheres for pairs YPQ − YPQ before and after segregation.
Before segregation
After segregation
1st sphere
2nd sphere
1st sphere
2 nd sphere
8Y
1.43 ± 0.09
0.71 ± 0.10
2.61 ± 0.11
1.02 ± 0.12
12Y
2.05 ± 0.11
1.12 ± 0.09
2.87 ± 0.13
1.39 ± 0.11
15Y
2.54 ± 0.09
1.25 ± 0.10
3.45 ± 0.11
1.54 ± 0.10
18Y
2.96 ± 0.12
1.60 ± 0.08
3.72 ± 0.10
1.82 ± 0.13
16
Table 3: Activation energies (in eV) for bulk diffusion calculated before and after segregation. Before segregation
After segregation
8Y
0.71 ± 0.04
0.88 ± 0.03
12Y
0.80 ± 0.02
0.95 ± 0.03
15Y
0.88 ± 0.03
1.01 ± 0.04
18Y
1.03 ± 0.03
1.04 ± 0.02
17
RESEARCH HIGHLIGHTS FOR
YTTRIUM SEGREGATION AND OXYGEN DIFFUSION ALONG
HIGH-SYMMETRY GRAIN BOUNDARIES IN YSZ
by Robert L. González-Romero and Juan J. Meléndez
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A study of yttrium segregation to three grain boundaries (GB) in YSZ is performed.
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A steady concentration of yttrium is reached after a short transient regime.
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Segregation does not change the coefficients for oxygen diffusion along the GB.
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The main effect after segregation appears for oxygen bulk diffusion.
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The effect is related to the rearrangement of oxygen vacancies after segregation.
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