Lattice Parameters-Kim

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Unformatted text preview: journal J . A m . Ceram. Soc., 72 [a] 1415-21 (1989) Lattice Parameters, Ionic Conductivities, and Solubility limits in Fluorite-Structure M02Oxide [M = Hf4+,Zr4+, Ce4+, Th4+,U4+)Solid Solutions Dae-Joon Kim*,* Department of Materials Science and Engineering, The University of Michigan, Ann Arbor, Michigan 48109 Changes in the lattice parameters of fluorite-type MOz oxides cation and host ~ a t i o n , " . ' ~cannot be rationalized simply by -'~ consideration of the ionic size difference alone. Inconsistencies (M = Hf4+,Zr4+,Ce4+,Th4+, U 4 + ) due to the formation of solid solutions can be predicted by proposed empirical can be found in the systems Ce0,-alkaline-earth oxide,' Tho,lanthanide oxide (Ln203),14~1s Zr0,-alkaline-earth oxide. I' and equations. The equations show the generalized relationship Accordingly, it is necessary to find more general criteria for the between dopant size and ionic conductivity in the binary prediction of the conductivities and the solubility limits in the systems of these oxides, illustrating that the smaller the difference between the dopant ionic radius and the critical dopant fluorite-structure M 0 2 oxide solid solutions. In the present paper, empirical equations are proposed which radius, the higher the conductivity. The solubility limit of the predict the changes in the lattice parameters of fluorite-structure same periodic group elements in fluorite-structure M 0 2 oxides decreases linearly with the square of Vegard's slope for each MO, oxide solid solutions as a function of dopant concentration. These equations were utilized to explain the role of dopant size in solute as determined from the proposed equations. [Key words: determining ionic conductivity and to clarify the correlation beoxides, lattice, ionic conduction, solid solutions, fluorite.] I. Introduction tween the solubility limit of dopants and crystal chemical parameters in the fluorite-type MO, oxide solid solutions. 11. M 0 2 oxides of IVB group elements, except titanium, and the actinides crystallize with the fluorite structure. The solid solutions of these oxides containing lower-valent cations are well-known ionic conductors frequently utilized as oxygen sensors, electrochemical oxygen pumps, and fuel cells.'.' The ionic conductivity of the fluorite oxides depends on the type of dopant and its c~ncentration.~-~ Maximum conductivity is achieved by alloying with dopants which cause very little expansion or contraction of the fluorite lattice.'.'" Thus, in the study of oxygen-ion conductors, it is desirable to determine the effect of dopants on the lattice parameters and their solubility limits in the cubic fluoritestructure phase. The changes in the lattice parameters of fluorite oxide solid solutions have been predicted from the consideration of the influence of dopants on the geometry of the fluorite lattice."-'3 However, the models which lead to these predictions have been applied to only two fluorite oxide systems, HfOzll and Zr02.11-13 applicaThe bility of such models to other fluorite oxide systems remains uncertain since, unlike HfOz and ZrO,, the other M 0 2 fluorite oxides such as Ce02, Thoz, and UOz can retain the cubic fluoritestructure at room temperature without the formation of a solid solution. Furthermore, the agreement of calculated lattice parameters from these models with measured values has not been accurate enough for the prediction of dopant size dependence of ionic conductivity of fluorite-structure oxides. For a given dopant valency (usually, di- or trivalent) and concentration, the oxygen-ion conductivity of fluorite oxides varies with the dopant size.3-6 It has been explained thst the effect of dopant cation size on conductivity is related to the effect on the dopant-oxygen-ion vacancy association energy, which governs the activation enthalpy for conduction, so that a dopant which minimizes the ionic size mismatch between host and dopant maximizes the c o n d u ~ t i v i t y . ~ . ~ - ' ~ However, this generalization of the dopant size effect on conductivity is not adequate to explain the conductivities, especially, in the systems CeO2-M20; and Ce02-alkaline-earth oxide.6 Also, the solubility limits of the same periodic group elements in the fluorite MOz oxides, which have been related to the ionic size difference between the dopant HE N . J. Dudney-contributing editor Manuscript No. 198916. Received August 30, 1988; approved December 13, 1988. Member, American Ceramic Society. Now with Oak Ridge National Laboratory, Oak Ridge, TN 37831-6069. T Procedure and Discussion (1) Equations for Lattice Parameters In the fluorite structure, cations are in eightfold coordination with their nearest neighbors and each anion is surrounded tetrahedrally by four cations. The solid solution of the fluorite-type M 0 2 oxides is obtained by the substitution of solute cations for host cations. Doping the fluorite oxide with aliovalent cations, whose valency is smaller than that of the host cation, creates oxygen-ion vacancies to achieve electrical neutrality in the substituted fluorite lattice. The changes in the lattice constants of the substitutional solid solution, particularly in the system Zr0,Y,O,, have been estimated by Aleksandrov et a / . and further elaborated by Ingel and Lewis,13 who considered the effect of dopant cation size on the geometry of the fluorite structure unit cell. Nevertheless, their model is not applicable to other fluoritetype MOz oxide systems, since they did not precisely account for the influence of the oxygen-ion vacancies created by Y201doping on the lattice geometry, but rather used somewhat arbitrary values of ionic sizes to compensate for the effect of vacancy formation. Glushkova et a / .I ' have taken account of the influence of both the radius difference and anion vacancy formation on the change of the lattice parameter. The contraction of the lattice due to the creation of oxygen-ion vacancies is incorporated through the consideration of the effective oxygen-ion radius, which becomes smaller as the dopant concentration increases. According to Glushkova et a / ., the effective radius varies as 0.138( 1 - rn/400)1'3 nm, where 0.138 nm is the radius of 0,- in fourfold coordination, rn is the mole percent of the lanthanide oxide in the form of LnO, 5 , and 3.6 is the correction factor for the association of vacancies. The lattice constants of ZrOz- and Hf02-Ln203 systems calculated using this model show agreement with the experimental values to within an error of 20.003 nm. This calculation results in greater deviations from the measured lattice parameters in Tho2and Ce0,-Ln,03 systems, since the correction factor of 3 . 6 is not valid in these systems. The deviations become even more pronounced with alkaline-earth dopants because of their smaller cation valency (2+) compared to the lanthanide elements (3+). Assuming Vegard's law to be valid, then a linear relationship exists between the solute concentration and the lattice constants of the resulting solid solutions. Multiple regression analyses were nerformed to correlate the changes in the lattice constants of fluo- ' 1415 1416 Journal of the American Ceramic Society-Kim Vol. 72, No. 8 rite-structure M 0 2 solid solutions with the differences in crystal chemical parameters of constituent cations. In these analyses, the differences in ionic radius, valency, and electronegativity of the dopant and host cations were multiplied by dopant concentration and used as independent variables. A similar procedure has been applied to predict the shape change of silicate single chains from the chemical composition of the silicate." The results of the regression analyses showed that the electronegativity parameter did not add to the prediction of lattice parameter changes. This is most likely due to the fact that the electronegativity is a function of the radius and valency." The empirical equations for the lattice parameter changes of fluorite oxide solid solutions, then, can be expressed as follows: dHf= 0.5098 + k (0.0203Ark + 0.00022Azk)mk (1) dc, = 0.5413 + 2 (0.0220Ark + 0.00015Azk)mk k (3) (4) dTh= 0.5596 + 2 (0.0212Ark + O.OOO1lAzk)m, k where d (in nanometers) is the lattice constant of the fluorite oxide solid solution at room temperature, Ark (in nanometers) is the difference in ionic radius (rk - rh) of the kth dopant (rk)and the host cation (rh) in eightfold coordination from Shannon's compilation,'' Azk is the valency difference (zk - zh), and mk is the mole percent of the kth dopant in the form of MO,, which can be represented by mk = nk Mk 100 + z ( n k - 1)M, k x 100 (5) where n k is the number of cations in the kth solute oxide and Mk is the mole percent of the kth dopant oxide (e.g., nk = 2 and Mk = 3 for doping with 3 mol% Y203). Table 1 shows a summary of the regression analyses. Equations (1) through (4)may be used to predict the lattice parameters of fluorite-structure M 0 2 oxide solid solutions. Figures 1 through 4 compare the calculated values of the lattice parameters from the equations with the measured values from the literature. The maximum deviation of the calculated lattice constants from the measured values is 0.0014 nm. In Figs. 1 and 4, the measured lattice constants are the same values used for the regression analyses. In Figs. 2 and 3, the measured values are adapted from Etsell and Flengas' and Kudo and Obayashi,j' respectively. The excellent agreement between the measured and calculated parameters shown in Figs. 1 through 4 allows the estimation of hypothetical room-temperature lattice constants of pure fluorite-structure Hf02 and Zr02, which are not stable at room temperature, using Eqs. (1) and (2), respectively. These equations have been successfully applied to calculate lattice parameters in the ternary systems of Hf02, Zr02, and CeO,, and the same type of equations have been utilized to predict the alloying effect on the changes in c / a axial ratio of tetragonal Zr02 solid solutions which are a strained form of the fluorite structure.36 From Eqs. (1) through (4), the critical ionic radius of the dopant, rcr in binary systems of fluorite-structure M 0 2 oxides can be obtained. Here, r, corresponds to the ionic radius of the dopant whose substitution for the host cation causes neither expansion nor contraction of the fluorite lattice. Thus, r, for a given dopant cation valency is calculated simply by setting to zero the term corresponding to the slope in each equation. Doping with a solute whose ionic radius is smaller than the critical value in a fluorite oxide will contract the fluorite lattice and vice versa. The existence of rr has not been well understood, especially in the systems Zr02-Mg027 and Zr02-Scz03.3The lattice = constant of the fluorite-structure Zr02 (rZr4+ 0.084 nm) stabi= lized by MgO (rMgz+ 0.089 nm) or Scz03(rsc1+ = 0.087 nm) decreases with increasing solute content, even though the ionic radii of the solute cations are larger than the Zr4+ ion. The decrease in the lattice constants in these systems occurs because rMg2+ and rsc3+ are smaller than r, for a divalent solute (0.1057 nm) and a trivalent solute (0.0948 nm) in Zr02, respectively, which is determined from Eq. (2). The value of r, for various valencies from Eqs. ( 1 ) through (4) are related to the host cation size in Fig. 5 . As shown, r r increases linearly with increasing ionic radius of the host cation and decreasing dopant valency. U 0 2 , whose solid solutions are important to nuclear fuel technology, also crystallizes with the fluorite structure. The stoichiometry of U02 solid solutions, which determines important physical properties such as thermal conductivit can be estimated by the measurement of lattice parameter~.~'iheempirical equation to calculate the lattice constants in U 0 2 solid solutions cannot be obtained by the same procedure employed for Eqs. (1) Table I. Fluorite oxide Dopant oxide Summary of Regression Analyses Ref. Degrees of freedom* R2 (%): Std error (nm) 20 HfOl 24 25, 26 39 99 0.0005 21 97 0.0007 Ce02 16 98 0.0003 30, 31 32 SrO Tho2 24 96 0.0003 uo2 35 *Total number of d values used in regression - 1 . 'Percentage of total variation about the mean value of d expressed by regression. August 1989 Lattice Parameters, Ionic Conductivities, Solubility Limits in MO, Oxide Solid Solutions 0.520 1417 E 4-4 c 8 0.520 0.516 d SlOl c v) 01 0.516 c 0 0 0 .c m 4-4 a , 0.512 0.512 - / CI Pr200 0 0 M A Smfl3 a0 23 mp3 DYZQ m I . , . , . , . w , . Ybfi *203 0.500 0.504 0.508 0.512 0.516 0.520 0. 4 0.508 0.508 0.512 0.516 0 20 Calculated lattice constant, nm Fig. 1. Comparison of measured and calculated lattice parameters of fluorite-structure Hf02 solid solutions. The measured values were obtained from references listed in Table I. Calculated lattice constant, nm Fig. 2. Comparison of measured and calculated lattice parameters of fluorite-structure ZrO, solid solutions. The measured values, obtained from Ref. 1, are for 10 mol% M203-ZrOZ. through (4), since U 0 2 is readily oxidized at elevated temperature so that the lattice constants of UOz solid solutions, which are heat-treated in a reducing or inert atmosphere, represent the effect of various degrees of nonstoi~hiometry.~"~"such a case, In the lattice parameter changes, due to the formation of anion vacancies, cannot be estimated simply from the dopant valency and concentration in the U 0 2 solid solutions. A formulation of the predictive equation for the lattice constants of U 0 2 solid solutions is described below. As discussed earlier, the effective anionic radius in the fluorite lattice due to the formation of oxygen-ion vacancies decreases with an increase in the number of vacancies. The decrease in the radius depends on the nature of the interaction of vacancies, so that the association of vacancies causes a smaller decrease in the effective oxygen-ion radius than does the noninteracting vacancies." Thus, the coefficient of AZ in Eqs. (1) through (4) is likely to represent the influence of the interactions on the contraction of each fluorite oxide lattice. It is found that the coefficient of Az is linearly related to the cation radius of the corresponding fluorite oxide. In Fig. 6, the coefficient decreases as the ionic radius of the host cation increases. This linear correlation, along with the relationships shown in Fig. 5, is utilized to define the equation for the lattice constants of fluorite-type U 0 2 solid solutions. The ionic radius of U4+ in eightfold coordination is 0.100 nm. From Fig. 6 , the coefficient of Az corresponding to the radius of 0.100 nm is estimated to be 0.000 13; then, the coefficient of Ar for UOz can be calculated from the relationship between r, and the host cation radius in Fig. 5 . To illustrate, r, for a valency of 3+ in U 0 2 is determined to be 0,1063 nm. From the definition of r < , t h e c o e f f i c i e n t of Ar is estimated by 0.00013/(0.1063 - 0.100). Knowing the pure UOz lattice parameter of 0.5468 nm,41the equation for the lattice constant of a UOz solid solution can be expressed as du = 0.5468 + z(0.0206Ark + 0.00013Azk)m~ k (6) In Fig. 7, the lattice parameters calculated by using Eq. ( 6 ) are compared with the literature values for the systems U 0 2 - T h 0 ~ 2 and -Ce02? where no oxygen-ion vacancy is created due to the formation of solid solutions and Az, = 0. The maximum deviation of the calculated lattice constants from the measured values is only 0.0010 nm. The remarkable agreement supports the validity of the correlations in Figs. 5 and 6 and of Eqs. (1) through (4), from which the relationships were derived. c v) / 0.544 c 0 0 V .c m - a , - 73 v) ?! 3 m a , 0.540 5 0.536 0.540 0.544 0.548 0.552 0.552 0.556 0.560 0. 64 Calculated lattice constant, nm Fig. 3. Comparison of measured and calculated lattice parameters of fluorite-structure CeO, solid solutions. The measured values, obtained from Ref. 31, are for 30 mol% MO, ,-CeO,. Calculated lattice constant, nm Fig. 4. Comparison of measured and calculated lattice parameters of fluorite-structure ThoLsolid solutions. The measured values were obtained from references listed in Table I. 1418 Journal of the American Ceramic Society-Kim Vol. 72, No. 8 (2) tonic Conductivity Ionic conduction in the fluorite-structure M0, oxides occurs by the migration of oxygen ions via a vacancy mechanism. The concentration of the oxygen-ion vacancies is increased by the formation of solid solutions with di- or trivalent dopant cations so that the ionic conductivity becomes enhanced. In most instances, the dopant cations and oxygen-ion vacancies form defect com2.4.7- 1 0 The ionic conductivity in these materials, u, can be expressed by an equation of the form u T = A exp(-H/kT) (7) Ir, - r d I . The ionic radius of Ce4+ in eightfold coordination is 0.097 nm and rc for a trivalent dopant in CeO, is calculated to be 0.1038 nm from Eq. (3). As shown in Table 11, the conductivity of CeO, solid solutions is not dependent upon ITh - r d ( ,but lrc - rdl. This generalization is also valid for the system Ce02alkaline-earth oxide, where rc is determined to be 0.1106 nm. The order of ionic conductivity of CeO, doped with the alkalineearth oxides is predicted to be CaO(0.112) > SrO(O.126) > MgO(0.089) > BaO(0.142) where T is the absolute temperature, A is a preexponential constant, and H is the activation enthalpy for conduction. Thus, at a given temperature, the conductivity increases with increasing A and decreasing H . The activation enthalpy is expressed as H = H, f (nf 2)H, (8) where the ionic radius (nanometers) of each cation in eightfold coordination is shown in parentheses. This order, within their solubility limits in the fluorite-structure phase, is consistent with the conductivities measured by Yahiro et al. It should be noted that MgO causes the smallest Ir,, - r,l among the alkaline-earth oxides considered. In the system Zr0,-M203, which has been investigated by Strickler and C a r l ~ o n the order of the ionic con,~ ductivity for the dopants considered is expected to be Yb203(0.0985)> Y2O3(O.1019) > S ~ ~ O ~ ( 0 . 0 8 7 0 ) where H , is the enthalpy for migration of a vacancy created by doping with aliovalent cations, n equals 1 and 2 for di- and trivalent dopants, respectively, and Ha is the enthalpy for association of a defect complex which is governed by the elastic strain field around the defect complex.'^'" At a given temperature the conductivity of fluorite-structure oxides is determined mainly by the H, term, which is known to be related to the ionic size mismatch between the host and dopant cations (rh - rd).4.x-'0 a result, As the effect of different dopants on the ionic conductivity of the fluorite-type M 0 2 oxides has been interpreted in the past solely by r, - r, so that a dopant which minimizes I,, - rd maximizes the c o n d u ~ t i v i t y . ~ . ~ ~ ~ ~ ' ~ this generalization is not Nevertheless, sufficient to explain the effects of di- and trivalent dopants on the ionic conductivity of Ce02.5.6 this respect, Kilner' suggested In the lattice parameter measurements to predict the effect of dopants on the ionic conductivity. From the earlier discussion of the significance of rr in determining lattice parameter, a change in the fluorite-structure lattice due to the substitution of a dopant cation in binary systems is determined not by rh - r, but by the difference between r, and the dopant ionic radius, r, - r d , which is likely to control the relaxation of the lattice in the vicinity of the dopant-oxygen-ion vacancy defect complex. Applicability of the proposed empirical equations to the prediction of the ionic conductivity is examined below. Table I1 illustrates the relationship between the bulk ionic conductivity of Ce02 doped with trivalent cations, which was determined by Gerhardt and Nowick,' and the absolute value of > Sm203(0. 1079) Again, the ionic radii (nanometers) of dopant cations in eightfold coordination are shown in the parentheses. In this system, r,. is calculated to be 0.0948 nm from Eq. (2). This prediction is in line with the experimental values by Strickler and Carlson with the exception of S c 2 0 3doping, which has been considered to lead to the presence of a second phase or a distorted structure within the composition range where the conductivities were measured. As illustrated above, the empirical equations are useful for predicting a dopant which optimizes the ionic conductivity of a fluorite oxide according to Kilner.' Besides a better prediction of the lattice parameters, an advantage in the use of these equations over the lattice parameter maps, proposed by Kilner' for this purpose, is the applicability to the systems which contain more than one dopant. Figure 8 shows the relationship between the activation energy for conduction in the system ZrO2-Th0,-Y2O3, which is determined by Rothman et al.,44and the deviation of lattice parameter from that of pure Zr02 due to the formation of solid solution, which can be calculated from Eq. (2). The calculated lattice parameters, which are not obtainable from the lattice parameter maps, are in good agreement with the measured values by Rothman et al., as shown in Table 111, where rn4+ and ry3+ are 0.105 and 0.1019 nm, respectively. The relationship in Fig. 8 ' 0.00025 I 1 4 N L 0 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.08 0.09 0.10 0.1 1 Ionic radius of host cation, nrn Fig. 5. Critical dopant radius, rc, as a function of host cation radius for dopants of several different valencies. Ionic radius of host cation, nrn Fig. 6. Relationship between the coefficient of AZ in Eqs. (1) through (4) and the host cation size. August 1989 Lattice Parameters, Ionic Conductivities, Solubility Limits in MO, Oxide Solid Solutions 1419 Table 11. Comparison of the Relationship between the Critical Dopant (rc),Dopant (rd),and Host Cation (rh)Radii and the Ionic Conductivity for Different Cations in CeOl Solid Solution ~ Dopant lrc (nm) - 'dl lh ' (nm) - 'dl (xi0-4(cn.cm)-')* Gd'+ 0.0015 Y3+ 0.0019 ~ a ~ + 0.0122 0.0083 0.0049 0.0190 3.2 2.6 1.9 *The values of u are the bulk ionic conductivity at 325C from 6 mol% M @ CeOz sintered for 4 h in Ref. 5. is in accord with the proposal that the maximization of conductivity is achieved by alloying with dopants which minimize the variation of the fluorite lattice.' It should be noted in Table 111 that the activation energy in this system is not controlled solely by a concentration of oxygen-ion vacancies created by doping with Y3+. diction from consideration of ionic sizes alone, by which MgO is expected to have the minimum value since the radius of Mg'+ is closest to that of Z 4 ' among the alkaline-earth elements. The same contradiction can be found in the system CeOz-alkalineearth oxide.' An alternative analysis, which reveals a consistent relationship between the dopant size and the solubility limit in the fluorite-structure oxides, is presented below. For a substitutional solid solution, the extent of solid solubility of various elements in a given host crystal lattice can be expressed as a function of the radius difference between the solute and host element and the valency of the solute.18 When a cation whose radius and valency are different from those of the host cation is introduced into the fluorite lattice, it creates a strain in the lattice unless the influences of the two factors cancel each other out. The elastic energy in the strained lattice governs the extent of the solid solution in such a way that the smaller the energy required to introduce a dopant cation, the wider the extent of solid solubility of the dopant. The elastic energy, W, in a substitutional solid solution can be written asI8 (3) Extent of Solubility Keller et al. l5 have suggested that the solubility limit of lanthanide oxides in T h o 2 increases as the absolute value of the difference (rTh4a- rLny+l decreases. When the solubility limit determined by Keller et al. is plotted as a function of the ionic radius of the lanthanide element, as shown in Fig. 9(A), the extent of solubility increases as the ionic size becomes larger. After reaching a maximum solubility in the vicinity of the radius of Nd3+ (0.1109 nm), the solubility of even larger elements decreases. Interestingly, the maximum in the solubility curve occurs for a solute ion with an ionic radius larger than that of Th4+ (0.105 nm), so that the trend of the solubilities in the system Th02-Ln203 cannot be rationalized by consideration of solute cation radius alone. Several attempts to correlate the solubility with crystal chemical parameters in Th02-Lnz03 can be found in the work of Diness and Roy.14 The same tendency of the maximum for a species whose ionic radius is larger than that of the host cation has been reported in fluorite-type Zr02 systems." Figure 9(B) shows the plots of heat of solution of alkaline-earth oxides in fluorite-structure Z r 0 2 , which were calculated by Mackrodt and Woodrow," as a function of the solute cation radius. The plots have a minimum enthalpy at the radius of Ca2+ for both the case of zero defect association and complete defect association of the dopant cation with the oxygen-ion vacancies. Again, this is contrary to the pre- w = 7Gv,(; 2 - I)* (9) where G is the shear modulus and V, and V are the molar volumes of a pure metal and its solid solution, respectively. For fluorite-type oxide solid solutions, Eq. (9) can be rearranged as 2 where do is the lattice constant of a pure host oxide and Ad is the Table 111. Composition, Activation Energy for Conduction, and Measured and Calculated Lattice Parameters of Fluorite Structure in the System ZrOt-ThOZ-YzO3 Composition (mol%) Tho, Y,O, Activation energy* (kJ.mol-') Lattice parameter (nm) Measured* Calculated' 18.5 17.0 8.6 5.0 0.0 3.5 1.0 4.4 3.0 9.0 119 112 97 92 84 0.52188 0.52058 0.5173 0.5148 0.5142 0.52097 0.51979 0.5164 0.5150 0.5145 *From Ref. 44. 'Using Eqs. (2) and (5) 120 , w a , J - g . Y 2 . 110- 9 a , c 100- a Y 80 0.002 ! I I 0.004 0.006 0.008 0. 1 0 Calculated lattice constant, nm Fig. 7. Comparison of measured and calculated lattice parameters of fluorite-stmcture U 0 2 solid solutions. The measured values of U02Tho, and -CeO, were obtained from Refs. 42 and 43, respectively. Deviation in lattice parameter, nm Fig. 8. Activation energy for conduction from Ref. 44 as a function of deviation in fluorite lattice parameter from hypothetical pure Zr02 as determined from Eqs. (2) and ( 5 ) . 1420 70 Journal of the American Ceramic Society-Kim 500 Vol 72, No. 8 60 - . Y 400 50 E 3 3 - - + completedefect association zero defect association 40 c0 .+0 (0 300 30 Th I I I c 200 ca Zr 0 20 10 a a I c 100- 0 0.09 0 - ' . , 0 0.08 0.09 . I . I . I . I . (6) , . 0.14 1 0.10 0.1 1 0.10 0.11 0.12 0.13 5 Ionic radius of solute cation, nrn Ionic radius o solute cation, nrn f Fig. 9. (A) Solubility limit of lanthanide oxides in fluorite structure Tho, at 1550C from Ref. 15 as a function of solute cation radius in eightfold coordination from Ref. 19. (B) Heat of solution of alkaline oxides in fluorite-structure ZrOz from Ref. 16 as a function of solute cation radius in eightfold coordination from Ref. 19. change in the lattice constant due to formation of a substitutional solid solution. Since Ad < do, the higher-order terms in the expansion of the cubic expression in Eq. (10) may be neglected, and the equation can be rewritten as W = 6GdoAd2 (11) Thus, in a fluorite-structure oxide solid solution, the relative solid solubility of elements in the same periodic group is expected to be linearly proportional to -Ad2. For a given solute concentration, the value of Ad is determined by Vegard's slope for each solute element, which corresponds to the slope of the linear relationship between the lattice parameter of a solid solution and the solute concentration. As can be seen in Eqs. (1) through (4) and (6), Vegard's slope contains two factors, Ar and Az , which affect the solute solubility as discussed above; therefore, it is appropriate to correlate the extent of solubility of a solute in a fluorite oxide with the square of its Vegard's slope. The solubility limits in Fig. 9(A) are replotted in Fig. 10(A) as a function of the squared Vegard's slope for lanthanide oxides in Thoz, calculated from Eq. (4). The plot in Fig. 10(A) shows a simple linear relationship between the squared Vegard's slope and the extent of solute solubility. There is no unexplained intermediate maximum solubility as was observed in the plot of the solubility as a function of solute ionic radius shown in Fig. 9(A). The Vegard's slope dependence of the solubility limit can be applied to ZrOz solid solutions as shown in Fig. 10(B), where the heat of solution of alkaline-earth oxide in the fluorite-structure ZrO, is plotted as a function of the square of Vegard's slope of the solute in ZrOz from Eq. (2). The same linear relationship as in Fig. 10(A) is obtained for both the zero and complete defect association cases. This is in contrast fo the plots in Fig. 9(B) of the enthalpies versus cation radius, which show unexplained minima in the heat of solution at the radius of CaZ+.It is evident from the correlations in the Figs. 10(A) and (B) that, in fluoritetype oxides, the relative solubility limit for the same periodic group elements can be predicted by the value of Vegard's slope for each element, which is determined by both radius and valency differences between the dopant and host cations. 111. Summary Equations for predicting the lattice parameters of fluorite-structure MO, oxide solid solutions were obtained from consideration of the effect of the solute ion size and oxygen-ion vacancies on the geometry of the fluorite unit cell. Applicability of these equations was examined and remarkable agreement between the calculated and measured lattice parameters was obtained in various fluorite- J D Y o \ 0 1 1 . , . , . , . , . , . I . 0 200 400 600 800 0 10 20 30 40 50 60 0 (Vegard's slope)2 x lo* (Vegard's slope)2 x 10' Fig. 10. (A) Solubility limit in Fig. 8(A) as a function of the square of Vegard's slope for the lanthanide elements in Tho, as determined from Eq. (4). (B) Heats of solution in Fig. 8(B) as a function of the square of Vegard's slope for the alkaline-earth elements in ZrO, as determined from Eq. (2). August 1989 Lattice Parameters, Ionic Conductivities, Solubility Limits in MO, Oxide Solid Solutions 1421 type M 0 2 oxide systems. It has been shown that a distortion of the fluorite lattice due to the formation of solid solutions in the binary systems is not determined by the mismatch between dopant and host cation radius but by the difference between dopant radius and the critical dopant cation radius ( r < ) which can be de, termined from the proposed empirical equations. 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