- Email: [email protected]
A defect-thermodynamic approach to PuO2−x and CeO2−x
Journal of Nuclear Materials 201 (1993) 17-26 North-Holland
A defect-thermodynamic
approach to PuO,_,
and CeO,_,
Akio ~akamura Japan Atomic Energy Research Institute, ChemistryDivision,Tokai-mura,Naka-gun, ibaraki 319-11, Japan
The prominent feature of the thermodynamic data of PuO,_, and CeO,_, is the strong sigmoidal variation of g(O,) and $0,) with X. In this paper, in search for a possible basic mechanism for the onset of such feature of &(O,) and $0,) in these hypostoichiometric fluorite-type oxides, a defect-therm~ynamic formulation is put forward based on the local Pu3+(Ce3+ )-V, interaction model in the mutual first nearest neighbor coordination sphere. Under some simplified assumptions on the V, arrangement on the oxygen sublattice the number of the local defect-defect configurations is reduced to 16, and using the virtual Gibbs formation energies of these 16 local configurations as the basic parameter the explicit analytical expressions are derived for their concentrations and for the thermodynamic quantities of oxygen (g(O*), x(0,) and S(0,)). The numerical evaluations show that such sigmoidal shape of both &(O,) and S(O,) as observed in these systems is interpreted and reproduced well by the present approach as a result of the coupled local defect-defect configurational changes between Pu(Ce) and 0 sites. Discussions are also made on the validity and the limitation of the present model.
1. Intr~uction In actinide series elements, as the main and common solid oxide phase the oxides with the fluorite structure are formed. Many of these actinide fluorite-
type oxides are widely nonstoichiometric such as UO,,, and (Pu, Am)O,_,. Following the. previous paper on UO,,, t11,in this paper the author presents a defectthermodynamic approach to PuO,_, and its rare-earth analog CeO,_,, in the hope of getting the more global scope of the defect structure and the thermodynamic properties of these highly defective fluorite oxides. (Due to the basic similarity of PuO,_, and CeO,_,, in below we mainly focus our discussion on CeO,_, for which more extensive experimental data are available.) The prominent feature of the thermodynamic data of the h~ostoichiometric CeO,_, and PuO,_, having oxygen vacancy (V,) type defects is the strong sigmoidal variation of h(O,) (and also of $0,)) with x 12-41 (see figs. 4 and 51, for UO,,, the h(O,) change is the more monotonous one with a shallow minimum around x = 0.23 [5]. Such strong x dependence of &(O,) means that the defect structure and therefore the defect formation enthalpy of the system as a whole are drastically changing with x. In the case of UOZcx we employed the quasi-chemical defect equilibria approach for the formation of several defect complexes in combination with the molar-volume dependent defect formation and interaction enthalpy terms [I]. Such an 0022-3115/93/$06.00
approach made so far by the present author is not enough to describe well the much pronounced sigmoidal x dependence of @O,) in the present hypostoichiometric systems. As for this point the same appears to be true for most of the various theoretical attempts reported in literature [6-141 spanning from the ~lution-thermodynamical, the statisticaI-dyn~ica1 to the atomistic lattice-dynamical ones (which are successful in their own many respects). A~ording~y in this paper, the author puts forward a defect-thermodynamic formulation to serve this purpose based on the local defect (Pu3+(Ce3+))-defect (V,) interaction model. Though based on some idealized assumptions and also phenomenological in nature, the present model can give a straightforward interpretation that such strong sigmoidal variation of both %(O,) and S(0,) as observed in these systems is indeed brought about as a direct result of the coupled confiurational changes of the strongly interacting local Pu3+(Ce3+)-V, configurations between Pu(Ce) and 0 sites with x and temperature T. 2. Description of the theoretical model 2.1. General background and basic ass~mp~ions of the model
We first discuss briefly the main motive for the present local defect-defect interaction model and
0 1993 - Elsevier Science Publishers B.V. All rights reserved
garding CeO, t iPu0, ~, i as the solid solution oi CeO,(PuO,) and Cc,O,(PuzO,), Kim’s equation givcb the following expressions for room temperature lattice parameters of the respective systems:
CeOzmx 0
c
-
Kim’s
eq.
+Q
=5.413 + 2(0.150 AZ + 2.20 Ar).r
for CcO, a(&
I
0
~
,
01
02 r
03
x
Fig. 1. Lattice parameter a,,(A) versus CeO,_, and PuO,_,. ~p~r~mental data: [lS] (0, A); PuO,_,, Markin and Rand 1171(v ). Empirical curves: Ce02__Y(-------); calculated in this study using Kim’s
*I relationship in GzO,_~~, Touzeiin [16j (O), Boivinea f, PuO,_, i- - empirical equation
[181. thereupon introduce several basic assumptions in formulating the theoretical model. ,Fig. 1 shows the experimental lattice parameters a(A) versus x plots for CeO,_, [IS] and PuOr_, [l&17] at high temperatures together with the emplricdl curves calculated for the respective systems according to the recent Kim’s proposal [18]. Kim’s proposal is originally made and successfully applied for the lattice parameter variations of various fluorite-type solid soiutions between the host MO&M = Hf”+, Zr’+, Cc4’, Th”.‘, U4+) and the aliovalent dopant oxides such as RO (R = Mg”, Ca2+, Sr2+, Ba*+) and Re,O, (Re = yli- sc3+ , trivalent rare earth cations). By properly incorporating the two basic parameters (Ar = rd - rh and Az = zd - zh; the ionic radii and charge diffcrences between the dopant (d) and the host (h) cations) Kim’s equations also reveals the decisive role of the local distortional effect due to the dopant-host cations’ size mismatch (Ar) in determining the solid solution properties such as the solub~iity limit of the dopant oxides and ionic conductivity of the system. When applied to these binary hypostoichiometric systems re-
, and
= 5.396 + Z(O.lSS AZ + 2.08 3r)s
for PuO, f. Inserting the ionic radii data (Cc” ’ : 0.97, Pul+ : 0.96. Cc3 + : 1.143, PLY’’ : 1.142) and converting to the high temperature values gives the cmpiricai curves of the respective systems shown in fig. 1, It is obvious that Kim’s equation gives a good description of the experimental lattice parameters of CeO, .., for which the more reliable and extensive data arc available. This result, together with the obvious fact that the dopant cation Ce3’ (Pu3’ ) in this case has an ionic radius almost comparable to the two largest trivalent rare-earth cations (La’” : 1.160, Pr3’ : 1.1261, strongly suggests that in these systems nonstoichiometry x causes relatively large local distortional effects around the cationic and anionic defects Ce’ ‘(Pu”~ 1 and V,,. This result also implies that Cc3 ‘(Pu” ) is troatcd basically as in the localized reduced charge state similar to other fixed valence trivalent rare-earth cations, and electron trapping V,(V:) type defect state may be excluded in these rare-earth and actinide fluorite-type oxides. Many intermediate ordered phases emerge at low temperature from the high temperature disordered nonstoichiometric CeO,_,(PuOZ_.,). as well as in other various solid solution systems mentioned above. In fig. 2 we show the two representative basic structural units for those intermediate ordered phases proposed in Iiterature [19,20]. In the latter model shown in fig. 21~ an unified structurai description of any intermediate ordered phase between CeZO, and CeOz is contemplated by combination of the two basic units (7F and 1) in variously ordered manner. Where the 7F unit iq coordinated Ce octahedra of composed of seven 80’ the stoichiometric fluorite structure, and the I unit is composed of central 60’- coordinated Ce octahedra with the (third N.N.) body-diagonal V,,-V,, configuration and the accompanying upper and lower six 70’ coordinated Cc octahedra. Note that the composition of 1 : 1 ordered structure of 7F(Ce,O,,) and l(Ce,O,,)
in the hexgonal form (y-phase Zri,Sc,O,,) corrcsponds well to that of the minimum of --h(O,I in Cc ,JOZh (X = 0.143 in CeO, ~.I as seen in fig. 4a. FinalIy at the limiting composition of x = 0.286 (G+O,,) to which the sharp drop of -x(0,) is ob-
A. Nakamura /A
19
defect-thermodynamic approach to PuO2 _ x and CeO, _ x
served, the system is entirely built up by I units, and this structure coincides with the V, string structure in the former model shown in fig. 2a. It is judged from both these models that the third N.N. body-diagonal V,-V, configuration (avoiding the first N.N. and the second N.N. V,-V, configurations) is preferred in these intermediate ordered phases and the similar tendency would still persist in the high-temperature disordered nonstoichiometric phase Ill]. Based on the above qualitative discussion we introduce here the next two basic assumptions of the present model: (1) The major part of the defect-structure and thermodynamic changes of these hypostoichiometric system with x is brought about by the changes of the strongly local configinteracting various Ce 3+(Ce4+)-V,(02-) urations with x in the mutual first N.N. coordination sphere (from the single Ce4+-O*- type one at x = 0). All the second N.N. (and the further distant) configurational correlations between Ce and oxygen (0) sites are neglected. Also due to the strong first and the second N.N. V,-V, repulsion, these V,-V, configurations are excluded.
(111)
I unit 7 F unit Fig. 2. (a) V, string structure along 111 direction proposed for Ce,O,, [19]. (b) Two basic structural units proposed for the fluorite-related intermediate ordered phase [201. Left: 7F unit composed of seven 802coordinated Ce octahedra (shaded cube). Right: I unit composed of one central 602coordinated Ce octahedra having body-diagonal 3rd N.N. V,-V, configuration (filled cube) and six neighbouring 70*- coordinated Ce octahedra (open cube).
0 : Ce
0 : Od
0 : OBG
Fig. 3. Designation of defect 0 site, O,, and background site, On,, in the fluorite-type lattice.
0
By this assumption we can greatly reduce the number of the local configurations (the chemical species) to the next 16 first N.N. types; Ce3+S4+(i) (i = 0, 1, 21, and V,(i), O’-(i) (i = 0, 1, 2, 3, 4), where i denotes the coordination number of V, around Ce3+S4+, and that of Ce3+ around V, and O*-, respectively. By assigning the virtual Gibbs formation energies (g) to these 16 local configurations (chemical species), we can designate the relative stability relationships between them in the fluorite phase. The next assumption is introduced in order to obtain the relatively simple expressions for the concentrations of these 16 local configurations which is the prerequisite for the formulation of x(0,). (2) The ordered V, arrangement on the parent fluorite 0 sublattice and the random distribution of V, on these defect 0 sites (0,) as depicted in fig. 3. That is, total 2 0 site is devided into 0.5 0, which can be occupied by V, and 1.5 O,, which cannot be occupied by V,, in a perfectly ordered manner. This assumption may clearly be too much an idealization of the actual situation in view of the basic “disordered” nature of the 0 sublattice of the original fluorite structure in which all the 0 sites are crystallographically equivalent. Boureau and Campserveux’s result under the same first and second N.N. V,-V, configuration exclusion is availabe for the concentrations of 0, and On, in such more realistic disordered case [ill. In contrast to the present assumption (2), their result shows the rapid decrease of 0, to zero beyond x = 0.26 in agreement with the observed for-
20
, and C’eO, ,
A. Nakamuru / A defect-thermodynamic approach to t’uOz
mation of Ce,O,, type ordered phase around .Y= 0.286. Then, the next step would be to find out some approximate expressions of the concentrations of these various local defect configurations (V,,(i), Ce3’(i), etc.) based on their result and thereupon derive the h(02) expression solvable on computer. The author’s attempt is now directed along this line, but so far not succcssful. So, the author here pursues the present model based on assumption (2) as an idealized alternative approach to the problem.
The total Gibbs energy G(total) of the system (per I mol of CeO,_,) is given by the sum of the following three component terms; G(tota1)
+ G(OV,)
+ G(O,,;).
(It))
The explicit expressions of the rcspcctive component terms arc given by the following equations using the (virtual) Gibbs formation energies (g) of the various chemical spccics: G(Ce)=
2.2. Formulation
= G(Ce)
i
{N~,[Ce”(i)]g(~e”(i))
, =o
of the model
+N,[Cc’-(i)]g(Cc’+(i)) In the and O’equations Cc(i) and given by [Cc(O)]=
simplest case of random distribution of on the 0, sites assumed above, all necessary to designate the concentrations Ce’+,+4(i) (i = 0, 1, 2) on the Ce sites
V,, the of are
-kT
lnN,[Ce(i)]!
/Nc,[Ce3+(i)]!N,[CeJ’(i)]!},
G(OYi) = C [Cc3+(0)]
+ [Ce4+(0)]
,
= 1-44x+4x’.
(11)
{NP,(i)lg(Y,(i))
0
+N(,[O,i ~(i)]g(O’
(i))
(1) --kT [Ce( I)] = [Cc’+(l)]
+ [Ce4+(1)]
=4x
[G(2)]
+ [Ce4+(2)]
=4x’,
- 8x’,
lnN..([V,(i)]
+ [Oi
iY,[V,,(i)l!N;,[O~~(i)] = [Ce3+(2)]
[Cc’+(O)]
+ [Cc’+(l)]
+ [Ce’+(2)]
=2x.
[o:-(i)]
=4C1(1 -,4)“+‘A’x, =4c;(l
-LQ4
(6)
where the coefficient A(B) for VJOi-) is the probability of finding Ce3+ on one Ce site around V,(Oz ) and is given by
A = (1-2x)[Ces+(l)]
B = (1 - 2x)[Ce3+(0)]
where N;, is Avogadro’s number. In the expressions of G(OV,) and G(O,,) we assume the same g(Oz-(i)) for both O:-(i) and O&(i). We also assume the (x and T independent) constant values of the virtual formation enthalpy (h) and entropy (.s) in g( = h -- Zs) for all the chemical species. From the internal equilibrium at constant x and r. i.e., aG(total)/a[Ce3+(i)] I .y.r’= 0 (i is either of 1. 2 and 3), we obtain the following two independent equations:
AG(OV,) 4
+ 2x[Ce”+(2)]
[CeWl ’
[CeCl>l
+ 2x[Ce”+(l)]
[Wl>l
[WO)l
(7)
for Ce’ t,4 + distribution present model.
- 2x)J around
O&
l, is random
[Cc'+(O)]
[Cc’+(l)]
(8)
AG(Ov, > 2
( 14)
[Cc’+(O)]
+RTIn[Ce4+(2)] where
=t”
+ AGCE34(0-2) [Ce”(2)]
the following
AGCE34(0-1) = 1.5 .,C,(2x)‘(l
+ AGCE34(0-1)
+RTIn[Cc4+(1)][Ce”+(0)]
For example, the coefficient A is given by the sum of two probability terms of finding Ce”(1) or CC”+(~) on one Ce site around V,, because Ce site around V, is either Ce(1) or Ce(2). Similarly, the concentration of O&(i) is given by [o;o(i)]
(12)
(4)
(5)
~‘B’(0.5 -X),
!),
(3)
where the brackets denote the concentration of the chemical species. The corresponding expression for the concentrations of V,(i) and Oi- (i) (i = O-4) is given by [V,(i)]
(i)])!
(2)
[Ce3+(0)]
abbreviations
= {g(Ces+(l))
=“’
(15)
are used: +g(Ce”+(O)))
(9) in the
-{s(Cc”+(O))
+g(Ce4+(l))}, (16)
21
A. Nakamura /A defect-thermodynamic approach to PuO, _ x and Ce02 _ x
AGCE34(0-2)
= (g(Ce3+(2))
+g(Ce4+(0))}
-{g(Ce3+(0))
+g(Ce4+(2))}. (17)
And introducing and O:-(i): A@,(i))
the following equations for both V,(i)
=g(V,(i))
Ag(O’-(i)) Ag(V,(i))
-g(V,(O)),
=g(O*-(i)) = Ag(V,(i)) +
Ag(O’-(i))
(18)
-g(O*-(O)),
(19)
- RT fn( [V,(i)]
-g(C),)/2
[wG)])/Pw)l~
= Ag(02-(i))
in terms of [Ce3’(l)] and X. Then, inserting these concentrations back to eq. (14) (or (15)) gives the implicit function of [Ce3’(1)], F([Ce3+(1))1, x, AG,) = 0, where AG, is the above Gibbs energy parameter. This is solved numerically for [Ce3’(1)] for a given x and AG, on computer using the successive bisection method. From eqs. (lo)-(14), we can obtain the following expression of g(O,) utilizing the relationship aG(total)/W,x I T = -g(O,)/2: = -g(O,)(Ce)/2
(20)
- RT ln([V,(i)]
+ [O:-W])/[O:-Wl
f
(21)
-
(23)
a02)(%3)/2~
where the explicit expressions terms are: -g(O2)(Ce)
we can express AG(OV,) in eqs. (14) and (15) as
- g(O,)(OV,)/2
=
of these component
AGCEA
2 AG(OV,)
=
cU(V&)) { i -
i.
. a[Y,(i)]/aA
&G(O’-(i)).
11,~ /x 1
a[@-]/W,T
[Ce(1)]4[Ce4+(0)]6 -RT
In [Ce(0)]4[Ce3+(O)]2[Ce4+(l)]4
1
,(O.L).
(22) Eqs. (14) and (15) lead to the coupling of the configurational disorder between Ce and 0 sublattices and show the way of this coupling at the present level of approximation. By solving numerically eqs. (14) and (1.5) on computer, we can determine the concentrations of all the chemical species for a given set of the Gibbs formation energy parameters; AG CE34(0- l), AGCE34(0-2), Ag(V,(i)) and Ag(O*-6)): In short, making 2 x eq. (14) -eq. (15) gives the equation containing only Ce 3+*4+(i) concentration terms. From the combination of this equation with eqs. (l)-(9), we can express all the concentrations of the chemical species
+8x
AGCEB i
[~e(~)l[~e(2)l[~e4+(1)12 -RT In[Ce4+(0)]
[Ce4+(2)][Ce(1)]*
(24)
-i302)(ovd)
2
Table 1
The Gibbs energy (g = h - Ts) parameters used to construct the theoretical g(OJ, %O,) and S(0,) curves in CeO,_, PuO,_,
in units of J/mol. AGCEOV
CeO,-, PUO, --x
CcO,_, PLlO,_,
PuO,_,
AGCE34(0-1)
-4.6
Ag(O’~(l))
Ag(O*-(2))
Ag(O*-(3))
-4.46x10”-(-1.46)T 2.62x lo3 - 1.07T
4.46 x lo3 - 1.46T - 4.71 X lo3 - ( - 2.67)T
A g(V,(2))
A g(V,(3N
5.35 x lo3 - 1.76T 6.28 x lo3 - 2.13T AsW,(lN
CeO,_,
AGCEB
393.8 x lo3 - 132.6T 451.9~ lo3 - 104.6T
2.51 x lo3 - 0.84T 1.26 x lo3 - 0.40T
0-OT x103-(-1.26)T
-2.51x103-(-0.84)T -8.79x103-(-2.78)T
13.05x lo3 -3.47T 20.92x lo3 - 6.28T
50.21 X lo3 - 16.747 56.48 x lo3 - 11.92T
AGCE34(0-2) -7.32x103-(-1.88)T -10.46x lo3 -(-5.02)T Ag(O*-(4)) -4.71X103-(-2.67)T -9.41X103-(--3.20)T A sW,(4N
- 62.76 x lo3 -( - 25.10)T -69.04x103-(-29.81)T
and
22
,
(4
CeO2-x
This
study
----
-
636OC
: ro8o”c ---____-_,ijooy
900
0 E .
2
j ‘i j
J
700 0
0.10
,
0 20
I
0
0 10
020
030 x
X
Fig. 4. ~ h(Oz) (a) and - .$O,) (b) versus x relationship in CeO,_,. (0) Bevan and Kordis [2], (0) Campserveux and Gerdanian [3], ( A 1 Panlener et al. [21]. For both X(0,) and S(Oz) theoretical curves obtained in this study are drawn at three temperatures.
This --.--
study
:
750°C
Fig. 5. - %Oz) (a) and - S(O?) (b) versus x relationship in PuO,_,. (0) Chereau et al. [4], (0) this study. For both x(02) and S(O,) theoretical curves obtained in this study are drawn at three temperatures.
A. Nakamura /A
-dW(%o) 2
defect-thermodynamic
a[OS)l aN=x
= cAg(02-(i)) i
AGCEOV
In eq. (24) we used the following abbreviations:
2g(Ce3+(0))
= AGCEA+g(V,(O)) = 2g(Ce3+(0))
-t 4g(Ce4+(1))
+g(V,(O))
- 6g( Ce4+(0)), AGCEB =g(Ce4+(2))
(27)
+g(Ce4+(0))
-2g(Ce4+(1)).
23
x and CeO, _ x
eqs. (24)-(26) as r’
(26) AGCEA=
approach to PuO,
(28)
We can also derive the explicit expression of h(O,) directly from eqs. (23)-(28) utilizing the relationship h(0,) = a(g(O,)/T)/$l/T) I X. As is apparent in eq. (24) for g(O,XCe), in these component terms of x(0,), the partial derivative of the concentrations of the various chemical species with respect to l/T at constant x appears. In short, we need to know the only two a[Ce3’(i>]/a(l/T)I X (i = 1, 2), and for these two, though somewhat lengthy and not given here, we can also derive the explicit analytical expressions from eqs. (14) and (1.5). Finally we obtain S(O,) by the relationship S(O,) = (h(O,) - g(O,))/T. For the numerical evaluations of the model, we need to adjust the total of 24 enthalpy (Ah) and entropy (AS) parameters (12 for each) in 12 Gibbs energy parameters (Ag) listed in table 1. Among these, the only one which is not defined yet by the above equations is AGCEOV. This quantitiy appears in eq. (23) for g(O,) by inserting the three component terms
-g(02-(0))
+ 4g(Ce4+(1))
- 6g(Ce4+(0))
-s(0’-(0)).
(29)
Eq. (29) means that AGCEOV is the Gibbs energy change of the fluorite crystal lattice due to the following defect formation reaction: 6Ce4+(0) + 02-(0) + 2Ce3+(0) + 4Ce4+(1) + V,(O) + O,(g)/2. Then, inspection of the Gibbs energy parameters in table 1 shows that we are simulating how and to what extent the actual defect formation process deviates from the simplest eq. (29) type to the more complex one involving more various kinds of defect species with x and T depending on the relative stability relationship (Ag) of these various defect species (and their various combinations) inside and between Ce and 0 sites.
3. Results and discussions All the Gibbs energy (g), enthalpy (h) and entropy (s) parameters we finally adopted are listed in table 1 for both CeO,_, and PuO,_,. The theoretical z(O,) and $0,) curves for CeO,_, and PuO,_, are shown in figs. 4 and 5, respectively. Those of g(O,)(= RTlnP& here PO*, in unit of MPa divided by the standard state oxygen pressure, 0.1013 MPa) for both
1
Pu o,_, 750°c
9oo'c
11oo'c
f5OO'C
16
-20
-10 Log
PO”
-20 (bl
Log
PO:
Fig. 6. logx versus logP& relationship in CeO,_, (a) and PuO,_, (b). ( -_) This study. (a) (0) Bevan and Kordis [2], (A ) Panlener et al. 1211,(0) Campserveux and Gerdanian [91. (b) (0) Markin and Rand 1161,(0) Swanson [22], (a) Atlas and Schlehman [231,(01 Serensen [24], ( v 1 Woodly [25].
A. Nakamura
24
/A
defect-thermodynamic
systems are shown in fig. 6 as logx versus logI’,:_ curves. Agreement with the reported experimental data in both systems appears to be satisfactory over the wide x and T range for all of these data except for x > 0.26 where experimental -h(O,) and -S(02) both seem to drop more sharply with X. Yet, the most prominent feature of sigmoidal h(0,) shape in CeO, ( is well reproduced by the present model (fig. 4a). For PuO,_, experimental data are rather sparse and their scatters are also relatively large. Chereua et al. [4] evaluated S(0,) of this system by the combination of their own x(0,) data (o in fig. Sa) with g(O:) data of ref. [16] (o in fig. 6b). As the latters seem to give somewhat higher g(O?), we reevaluated S(O,) using the average g(O?) data at 1100°C in fig. 6b. which are shown in fig. .5b as filled circles. The present thoretical curves closely follow these reevaluated S(O,) data. As seen in fig. 6, the present theoretical &O,) curves also reproduce the reported miscibility gap at low temperature and small x region in both systems. According to the present model, the sigmoidal shape of h(Oz) in CeO,_ r is brought about by the coupled complex local configurational changes between Cc3 +,4i on Ce sites and O,y- , V,, on 0 sites with x and T.
z
approach to PuO,
,
Therefore. such sigmoidal shape also appears in S(0,) (fig. 4b), but to the lesser extent due to the existence of the prevalent configurational entropy term itself which steadily shifts S(0,) to the lower value side. We can understand the present situation more clearly in fig. 7. which shows how the three component terms cooperate to produce such a coupled unique shape of both h(O,) and S(0,). Although the major part of the initial drop of --&(02) is due to the OBci term, its sigmoidal shape is mostly due to the internally coupled Ce and OV, terms, the relative contribution of the former Cc term being much greater in its final sharp drop beyond the peak around x = 0.24. The essential “structure” of the Gibbs energy paramcters in table 1 which causes such @O,) and S(OZ) shape and thereupon derived defect structure of these systems arc brictly summarized as follows: (1) very large positive and negative values of h and s of Ag(V(3)) and Ag(V,(4)) and also of themselves, rcspectively. These means that V,(3) and V,(4) are the least and the most favorable state for V,, respectively. For example, {Ag(V,(3)) + Ag(O’- (i))) - (Ag(V,(i) t Ag(O’-(3))} = (g(VJ3)) + g(O*-(iI) - (g(V,,(i)) + R (0’ (3))) have large positive values for any i(=
(
E \ 2
i --__ 0 \ -----..__.____ ----------.. _!6_____ .-.._ ___________ . . . . __ ‘. ‘.__
\
0” lJ= 4
, and CEO,
\
-1 O(
-___
ovd __________----------__I
I
-2013
I 636’C
-301
,
/ O'-
(4 Fig. 7. Relative
111
03
02
0.1 X
variations
of - L(02)
b)
(a) and
- .X0,)
(b) and their Y = 0.001.
component
terms
with
x. References
point:
(a) .r = 0. (h)
A. Nakamura 0, 1, 2, 4),
/ A defect-thermodynamic
which means that the defect equilibria, V,(i) + G:-(3) ti V,(3) + O;-(i), are shifted extremely to the left-hand side for any i avoiding always the formation of ~~(3). If their absolute values, I Ag(V,(3)) I and / Ag(V0(4))l are made smaller, the sigmoidal shape becomes weaker, which is seen in figs. 4 and 5 through their temperature dependence. (2) The combination of AGCE34(0-1) (> 0) (eq. (16)) and AGCE34(0-2) (< 0) (eq. (17)) so as to make the positive value of (2AGCE34(0-1) - AGCE34(0-2)). These means that Ce3+ favors Ce(2) the best, Ce(0) the next and Ce(1) the least, and also the defect equifibria, Ce3’(0) + Ce3+(2) + 2Ce4+(1) * Ce4+(0) + Ce4+(2) f 2Ce3+(l), is shifted to the left-hand side to some extent. The subtlety of these Gibbs energy parameter changes is illustrated by the considerably different shape of x(0,) between CeO,_, and PuO,_, in spite of the fact that the both systems share the common basic “structure” of the Gibbs energy parameters (1) and (2) as seen in table 1. Of course, due to the idealized assumptions adopted in fromulating the model, we need to receive the numerical details of these defects’ Gibbs energy parameters and therefrom derived defect structure of these systems with a grain of salt. As for the assumption (2) shown in fig. 3, the author is now attempting to reformulate the model on the more realistic disordered type treatment 1111, and this situation will become much clearer after some results are obtained from such an attempt. Yet, it is the author’s view that the basic idea of the present approach based on the local defect-defect interaction model and therefrom derived basic interpretation on the coupled strong sigmoidal variation of %(O,) and $0,) is a reasonable one. This view seems to be partly supported by the recent Monte Carlo type simulational results on CeO,_, reported by Boureau et al. 1141: Their results show that the long-range Coulomb interaction between the charged defect causes the decrease of %O,) in the relatively small x region (n < 0.15), but this decrease is approximately + (or less) of the experimental one shown in fig. 4a. For the deeper understanding of the defect structure and the thermodynamic properties of these highly defective actinide and related fluorite-type oxides involving both the local (short-range) and long-range interaction effects, much remains to be done in the future through the mutual interaction and cooperation between the Present type more phenomenological approach and the more fundamental defect-physical approach such as the above Monte Carlo [12-141 and the lattice-dynamical simulation techniques [26].
approach to FuOz _ x and CeO, _ x
25
4. Conclusion In this paper, the author proposed a defect-thermodynamic approach to CeO,_, and PuO,_, based on the first N.N. Ce3+*4+-V,(02-) interaction model. Under some simplified assumptions on V, arrangement on the oxygen sublattice, the number of local configurations (the chemical species) is reduced to 16, all their concentrations given by simple equations. Given their Gibbs energy parameters (g), we can derive the relatively simple analytical expression of g(O,), h(0,) and $0,) which are solved numerically on computer. With the proper choice of the g parameters we can reproduce well all the thermod~amic data of oxygen of these systems over the wide x and T range. The g parameters indicate that Ce3+(2), Ce3’(0) and Ce3’(1) are the most, the next and the least stable local states for Ce3’ respectively, and V,(3) and V,(4) are the urmost unstable and the urmost stable local states for V,, respectively. The onset of the sigmoidal x(0,) and $0,) in these systems is interpreted as a result of the coupled local configurational changes between Ce and 0 sites with x and T. Irrespective of the idealized and phenomenological nature of the model, the author believes that the basic idea of the present approach based on the local defect-defect interaction model is a reasonable one and combined with other various approach paves the way to our deeper understanding on the thermodynamic and defect properties of these actinide and related ~uorite-type oxides.
References [ll A. Nakamura and T. Fujino, J. Nucl. Mater. 167 (1989) 121 “,4;. Bevan and J. Kordis, J. Inorg. Nucl. Chem 26 (1964) 1509. 131J. Campserveux and P. Gerdanian, J. Chem. Thermodyn. 6 (1974) 795. 141 P. Chereau, G. Dean, M. De France and P. Gerdanian, J. Chem. Thermodyn. 9 (1977) 211. [5j A. Nakamura and T. Fujino, J. Nucl. Mater. 149 (1987) 80. [61 T.B. Lindemer, Calphad 10 (1986) 129. f71 M. Hillert and BO Jansson, J. Am. Ceram. Sot. 69 (1986) 732. [8l L.M. Atlas, J. Phys. Chem. SoIids 29 (1968) 91. 191J. Campserveux and P. Gerdanian, J. Solid State Chem. 23 (1978) 73. [lOI L. Manes, Nonstoichiometric Oxides, ed. T. Sorensen (Academic Press, New York, 1981) p.p. 99-154. 1111G. Boureau and J. Campserveux, Philos. Mag. 36 (1977) 9.
26
A. Nakamura
/A
defect-thermodynamic
1121 R. Tetot, M. Benzakour and G. Boureau. .I. Phys. Chem. Solids 48 (1988) 381. [13] G. Boureau and R. Tetot, Phys. Rev. B40 (1989) 2304 and 2311. [I41 R. Tetot, M. Benzakour and G. Boureau, J. Phys. Chem. Solids 51 (1990) 545. [15] B. Touzelin, J. Nucl. Mater. 101 (1981) 92. [16] T.L. Markin and M.H. Rand, Thermodynamics, vol. I (IAEA, Vienna, 1966) p. 145. [17] J.C. Boivineau, J. Nucl. Mater 60 (lY76) 31. [18] D.J. Kim, J. Am. Ceram. Sot. 72 (1989) 1415. [19] B.G. Hyde and L. Eyring, Rare Earth Research III. ed. L. Eyring (Gordon and Breach, New York, 1965) p. 623. [20] M.R. Thornber and D.J.M. Bevan, Acta Crystallogr. 824 (1968) 1183.
approach to PuO,
* and CeO,
,
[21] R.J. Panlener, R.N. Blumenthal and J.E. Carnier. J. Phys. Chem. Solids 36 (1975) 1213. [22] G.C. Swanson, Los Alamos Scientific Laboratory Report LA-6063-T ( 1975). [23] L.M. Atlas and G.J. Schlehman. Thermodynamics I1 (IAEA, Vienna, 1966) p. 407. [24] O.T. Sgrensen, Plutonium lY75 and Other Actinidcs, eds. J. Blank and R. Lindner (North-Holland, Amsterdam, 1976) p. 123. [2S] R.E. Woodly, J. Nucl. Mater. Y6 (1981) 5. [26] R.W. Grimes, C.R.A. Catlow and A.M. Stoneham. J. Chem. Sot. Trans II, 85 (1989) 335.
A defect-thermodynamic
approach to PuO,_,
and CeO,_,
Akio ~akamura Japan Atomic Energy Research Institute, ChemistryDivision,Tokai-mura,Naka-gun, ibaraki 319-11, Japan
The prominent feature of the thermodynamic data of PuO,_, and CeO,_, is the strong sigmoidal variation of g(O,) and $0,) with X. In this paper, in search for a possible basic mechanism for the onset of such feature of &(O,) and $0,) in these hypostoichiometric fluorite-type oxides, a defect-therm~ynamic formulation is put forward based on the local Pu3+(Ce3+ )-V, interaction model in the mutual first nearest neighbor coordination sphere. Under some simplified assumptions on the V, arrangement on the oxygen sublattice the number of the local defect-defect configurations is reduced to 16, and using the virtual Gibbs formation energies of these 16 local configurations as the basic parameter the explicit analytical expressions are derived for their concentrations and for the thermodynamic quantities of oxygen (g(O*), x(0,) and S(0,)). The numerical evaluations show that such sigmoidal shape of both &(O,) and S(O,) as observed in these systems is interpreted and reproduced well by the present approach as a result of the coupled local defect-defect configurational changes between Pu(Ce) and 0 sites. Discussions are also made on the validity and the limitation of the present model.
1. Intr~uction In actinide series elements, as the main and common solid oxide phase the oxides with the fluorite structure are formed. Many of these actinide fluorite-
type oxides are widely nonstoichiometric such as UO,,, and (Pu, Am)O,_,. Following the. previous paper on UO,,, t11,in this paper the author presents a defectthermodynamic approach to PuO,_, and its rare-earth analog CeO,_,, in the hope of getting the more global scope of the defect structure and the thermodynamic properties of these highly defective fluorite oxides. (Due to the basic similarity of PuO,_, and CeO,_,, in below we mainly focus our discussion on CeO,_, for which more extensive experimental data are available.) The prominent feature of the thermodynamic data of the h~ostoichiometric CeO,_, and PuO,_, having oxygen vacancy (V,) type defects is the strong sigmoidal variation of h(O,) (and also of $0,)) with x 12-41 (see figs. 4 and 51, for UO,,, the h(O,) change is the more monotonous one with a shallow minimum around x = 0.23 [5]. Such strong x dependence of &(O,) means that the defect structure and therefore the defect formation enthalpy of the system as a whole are drastically changing with x. In the case of UOZcx we employed the quasi-chemical defect equilibria approach for the formation of several defect complexes in combination with the molar-volume dependent defect formation and interaction enthalpy terms [I]. Such an 0022-3115/93/$06.00
approach made so far by the present author is not enough to describe well the much pronounced sigmoidal x dependence of @O,) in the present hypostoichiometric systems. As for this point the same appears to be true for most of the various theoretical attempts reported in literature [6-141 spanning from the ~lution-thermodynamical, the statisticaI-dyn~ica1 to the atomistic lattice-dynamical ones (which are successful in their own many respects). A~ording~y in this paper, the author puts forward a defect-thermodynamic formulation to serve this purpose based on the local defect (Pu3+(Ce3+))-defect (V,) interaction model. Though based on some idealized assumptions and also phenomenological in nature, the present model can give a straightforward interpretation that such strong sigmoidal variation of both %(O,) and S(0,) as observed in these systems is indeed brought about as a direct result of the coupled confiurational changes of the strongly interacting local Pu3+(Ce3+)-V, configurations between Pu(Ce) and 0 sites with x and temperature T. 2. Description of the theoretical model 2.1. General background and basic ass~mp~ions of the model
We first discuss briefly the main motive for the present local defect-defect interaction model and
0 1993 - Elsevier Science Publishers B.V. All rights reserved
garding CeO, t iPu0, ~, i as the solid solution oi CeO,(PuO,) and Cc,O,(PuzO,), Kim’s equation givcb the following expressions for room temperature lattice parameters of the respective systems:
CeOzmx 0
c
-
Kim’s
eq.
+Q
=5.413 + 2(0.150 AZ + 2.20 Ar).r
for CcO, a(&
I
0
~
,
01
02 r
03
x
Fig. 1. Lattice parameter a,,(A) versus CeO,_, and PuO,_,. ~p~r~mental data: [lS] (0, A); PuO,_,, Markin and Rand 1171(v ). Empirical curves: Ce02__Y(-------); calculated in this study using Kim’s
*I relationship in GzO,_~~, Touzeiin [16j (O), Boivinea f, PuO,_, i- - empirical equation
[181. thereupon introduce several basic assumptions in formulating the theoretical model. ,Fig. 1 shows the experimental lattice parameters a(A) versus x plots for CeO,_, [IS] and PuOr_, [l&17] at high temperatures together with the emplricdl curves calculated for the respective systems according to the recent Kim’s proposal [18]. Kim’s proposal is originally made and successfully applied for the lattice parameter variations of various fluorite-type solid soiutions between the host MO&M = Hf”+, Zr’+, Cc4’, Th”.‘, U4+) and the aliovalent dopant oxides such as RO (R = Mg”, Ca2+, Sr2+, Ba*+) and Re,O, (Re = yli- sc3+ , trivalent rare earth cations). By properly incorporating the two basic parameters (Ar = rd - rh and Az = zd - zh; the ionic radii and charge diffcrences between the dopant (d) and the host (h) cations) Kim’s equations also reveals the decisive role of the local distortional effect due to the dopant-host cations’ size mismatch (Ar) in determining the solid solution properties such as the solub~iity limit of the dopant oxides and ionic conductivity of the system. When applied to these binary hypostoichiometric systems re-
, and
= 5.396 + Z(O.lSS AZ + 2.08 3r)s
for PuO, f. Inserting the ionic radii data (Cc” ’ : 0.97, Pul+ : 0.96. Cc3 + : 1.143, PLY’’ : 1.142) and converting to the high temperature values gives the cmpiricai curves of the respective systems shown in fig. 1, It is obvious that Kim’s equation gives a good description of the experimental lattice parameters of CeO, .., for which the more reliable and extensive data arc available. This result, together with the obvious fact that the dopant cation Ce3’ (Pu3’ ) in this case has an ionic radius almost comparable to the two largest trivalent rare-earth cations (La’” : 1.160, Pr3’ : 1.1261, strongly suggests that in these systems nonstoichiometry x causes relatively large local distortional effects around the cationic and anionic defects Ce’ ‘(Pu”~ 1 and V,,. This result also implies that Cc3 ‘(Pu” ) is troatcd basically as in the localized reduced charge state similar to other fixed valence trivalent rare-earth cations, and electron trapping V,(V:) type defect state may be excluded in these rare-earth and actinide fluorite-type oxides. Many intermediate ordered phases emerge at low temperature from the high temperature disordered nonstoichiometric CeO,_,(PuOZ_.,). as well as in other various solid solution systems mentioned above. In fig. 2 we show the two representative basic structural units for those intermediate ordered phases proposed in Iiterature [19,20]. In the latter model shown in fig. 21~ an unified structurai description of any intermediate ordered phase between CeZO, and CeOz is contemplated by combination of the two basic units (7F and 1) in variously ordered manner. Where the 7F unit iq coordinated Ce octahedra of composed of seven 80’ the stoichiometric fluorite structure, and the I unit is composed of central 60’- coordinated Ce octahedra with the (third N.N.) body-diagonal V,,-V,, configuration and the accompanying upper and lower six 70’ coordinated Cc octahedra. Note that the composition of 1 : 1 ordered structure of 7F(Ce,O,,) and l(Ce,O,,)
in the hexgonal form (y-phase Zri,Sc,O,,) corrcsponds well to that of the minimum of --h(O,I in Cc ,JOZh (X = 0.143 in CeO, ~.I as seen in fig. 4a. FinalIy at the limiting composition of x = 0.286 (G+O,,) to which the sharp drop of -x(0,) is ob-
A. Nakamura /A
19
defect-thermodynamic approach to PuO2 _ x and CeO, _ x
served, the system is entirely built up by I units, and this structure coincides with the V, string structure in the former model shown in fig. 2a. It is judged from both these models that the third N.N. body-diagonal V,-V, configuration (avoiding the first N.N. and the second N.N. V,-V, configurations) is preferred in these intermediate ordered phases and the similar tendency would still persist in the high-temperature disordered nonstoichiometric phase Ill]. Based on the above qualitative discussion we introduce here the next two basic assumptions of the present model: (1) The major part of the defect-structure and thermodynamic changes of these hypostoichiometric system with x is brought about by the changes of the strongly local configinteracting various Ce 3+(Ce4+)-V,(02-) urations with x in the mutual first N.N. coordination sphere (from the single Ce4+-O*- type one at x = 0). All the second N.N. (and the further distant) configurational correlations between Ce and oxygen (0) sites are neglected. Also due to the strong first and the second N.N. V,-V, repulsion, these V,-V, configurations are excluded.
(111)
I unit 7 F unit Fig. 2. (a) V, string structure along 111 direction proposed for Ce,O,, [19]. (b) Two basic structural units proposed for the fluorite-related intermediate ordered phase [201. Left: 7F unit composed of seven 802coordinated Ce octahedra (shaded cube). Right: I unit composed of one central 602coordinated Ce octahedra having body-diagonal 3rd N.N. V,-V, configuration (filled cube) and six neighbouring 70*- coordinated Ce octahedra (open cube).
0 : Ce
0 : Od
0 : OBG
Fig. 3. Designation of defect 0 site, O,, and background site, On,, in the fluorite-type lattice.
0
By this assumption we can greatly reduce the number of the local configurations (the chemical species) to the next 16 first N.N. types; Ce3+S4+(i) (i = 0, 1, 21, and V,(i), O’-(i) (i = 0, 1, 2, 3, 4), where i denotes the coordination number of V, around Ce3+S4+, and that of Ce3+ around V, and O*-, respectively. By assigning the virtual Gibbs formation energies (g) to these 16 local configurations (chemical species), we can designate the relative stability relationships between them in the fluorite phase. The next assumption is introduced in order to obtain the relatively simple expressions for the concentrations of these 16 local configurations which is the prerequisite for the formulation of x(0,). (2) The ordered V, arrangement on the parent fluorite 0 sublattice and the random distribution of V, on these defect 0 sites (0,) as depicted in fig. 3. That is, total 2 0 site is devided into 0.5 0, which can be occupied by V, and 1.5 O,, which cannot be occupied by V,, in a perfectly ordered manner. This assumption may clearly be too much an idealization of the actual situation in view of the basic “disordered” nature of the 0 sublattice of the original fluorite structure in which all the 0 sites are crystallographically equivalent. Boureau and Campserveux’s result under the same first and second N.N. V,-V, configuration exclusion is availabe for the concentrations of 0, and On, in such more realistic disordered case [ill. In contrast to the present assumption (2), their result shows the rapid decrease of 0, to zero beyond x = 0.26 in agreement with the observed for-
20
, and C’eO, ,
A. Nakamuru / A defect-thermodynamic approach to t’uOz
mation of Ce,O,, type ordered phase around .Y= 0.286. Then, the next step would be to find out some approximate expressions of the concentrations of these various local defect configurations (V,,(i), Ce3’(i), etc.) based on their result and thereupon derive the h(02) expression solvable on computer. The author’s attempt is now directed along this line, but so far not succcssful. So, the author here pursues the present model based on assumption (2) as an idealized alternative approach to the problem.
The total Gibbs energy G(total) of the system (per I mol of CeO,_,) is given by the sum of the following three component terms; G(tota1)
+ G(OV,)
+ G(O,,;).
(It))
The explicit expressions of the rcspcctive component terms arc given by the following equations using the (virtual) Gibbs formation energies (g) of the various chemical spccics: G(Ce)=
2.2. Formulation
= G(Ce)
i
{N~,[Ce”(i)]g(~e”(i))
, =o
of the model
+N,[Cc’-(i)]g(Cc’+(i)) In the and O’equations Cc(i) and given by [Cc(O)]=
simplest case of random distribution of on the 0, sites assumed above, all necessary to designate the concentrations Ce’+,+4(i) (i = 0, 1, 2) on the Ce sites
V,, the of are
-kT
lnN,[Ce(i)]!
/Nc,[Ce3+(i)]!N,[CeJ’(i)]!},
G(OYi) = C [Cc3+(0)]
+ [Ce4+(0)]
,
= 1-44x+4x’.
(11)
{NP,(i)lg(Y,(i))
0
+N(,[O,i ~(i)]g(O’
(i))
(1) --kT [Ce( I)] = [Cc’+(l)]
+ [Ce4+(1)]
=4x
[G(2)]
+ [Ce4+(2)]
=4x’,
- 8x’,
lnN..([V,(i)]
+ [Oi
iY,[V,,(i)l!N;,[O~~(i)] = [Ce3+(2)]
[Cc’+(O)]
+ [Cc’+(l)]
+ [Ce’+(2)]
=2x.
[o:-(i)]
=4C1(1 -,4)“+‘A’x, =4c;(l
-LQ4
(6)
where the coefficient A(B) for VJOi-) is the probability of finding Ce3+ on one Ce site around V,(Oz ) and is given by
A = (1-2x)[Ces+(l)]
B = (1 - 2x)[Ce3+(0)]
where N;, is Avogadro’s number. In the expressions of G(OV,) and G(O,,) we assume the same g(Oz-(i)) for both O:-(i) and O&(i). We also assume the (x and T independent) constant values of the virtual formation enthalpy (h) and entropy (.s) in g( = h -- Zs) for all the chemical species. From the internal equilibrium at constant x and r. i.e., aG(total)/a[Ce3+(i)] I .y.r’= 0 (i is either of 1. 2 and 3), we obtain the following two independent equations:
AG(OV,) 4
+ 2x[Ce”+(2)]
[CeWl ’
[CeCl>l
+ 2x[Ce”+(l)]
[Wl>l
[WO)l
(7)
for Ce’ t,4 + distribution present model.
- 2x)J around
O&
l, is random
[Cc'+(O)]
[Cc’+(l)]
(8)
AG(Ov, > 2
( 14)
[Cc’+(O)]
+RTIn[Ce4+(2)] where
=t”
+ AGCE34(0-2) [Ce”(2)]
the following
AGCE34(0-1) = 1.5 .,C,(2x)‘(l
+ AGCE34(0-1)
+RTIn[Cc4+(1)][Ce”+(0)]
For example, the coefficient A is given by the sum of two probability terms of finding Ce”(1) or CC”+(~) on one Ce site around V,, because Ce site around V, is either Ce(1) or Ce(2). Similarly, the concentration of O&(i) is given by [o;o(i)]
(12)
(4)
(5)
~‘B’(0.5 -X),
!),
(3)
where the brackets denote the concentration of the chemical species. The corresponding expression for the concentrations of V,(i) and Oi- (i) (i = O-4) is given by [V,(i)]
(i)])!
(2)
[Ce3+(0)]
abbreviations
= {g(Ces+(l))
=“’
(15)
are used: +g(Ce”+(O)))
(9) in the
-{s(Cc”+(O))
+g(Ce4+(l))}, (16)
21
A. Nakamura /A defect-thermodynamic approach to PuO, _ x and Ce02 _ x
AGCE34(0-2)
= (g(Ce3+(2))
+g(Ce4+(0))}
-{g(Ce3+(0))
+g(Ce4+(2))}. (17)
And introducing and O:-(i): A@,(i))
the following equations for both V,(i)
=g(V,(i))
Ag(O’-(i)) Ag(V,(i))
-g(V,(O)),
=g(O*-(i)) = Ag(V,(i)) +
Ag(O’-(i))
(18)
-g(O*-(O)),
(19)
- RT fn( [V,(i)]
-g(C),)/2
[wG)])/Pw)l~
= Ag(02-(i))
in terms of [Ce3’(l)] and X. Then, inserting these concentrations back to eq. (14) (or (15)) gives the implicit function of [Ce3’(1)], F([Ce3+(1))1, x, AG,) = 0, where AG, is the above Gibbs energy parameter. This is solved numerically for [Ce3’(1)] for a given x and AG, on computer using the successive bisection method. From eqs. (lo)-(14), we can obtain the following expression of g(O,) utilizing the relationship aG(total)/W,x I T = -g(O,)/2: = -g(O,)(Ce)/2
(20)
- RT ln([V,(i)]
+ [O:-W])/[O:-Wl
f
(21)
-
(23)
a02)(%3)/2~
where the explicit expressions terms are: -g(O2)(Ce)
we can express AG(OV,) in eqs. (14) and (15) as
- g(O,)(OV,)/2
=
of these component
AGCEA
2 AG(OV,)
=
cU(V&)) { i -
i.
. a[Y,(i)]/aA
&G(O’-(i)).
11,~ /x 1
a[@-]/W,T
[Ce(1)]4[Ce4+(0)]6 -RT
In [Ce(0)]4[Ce3+(O)]2[Ce4+(l)]4
1
,(O.L).
(22) Eqs. (14) and (15) lead to the coupling of the configurational disorder between Ce and 0 sublattices and show the way of this coupling at the present level of approximation. By solving numerically eqs. (14) and (1.5) on computer, we can determine the concentrations of all the chemical species for a given set of the Gibbs formation energy parameters; AG CE34(0- l), AGCE34(0-2), Ag(V,(i)) and Ag(O*-6)): In short, making 2 x eq. (14) -eq. (15) gives the equation containing only Ce 3+*4+(i) concentration terms. From the combination of this equation with eqs. (l)-(9), we can express all the concentrations of the chemical species
+8x
AGCEB i
[~e(~)l[~e(2)l[~e4+(1)12 -RT In[Ce4+(0)]
[Ce4+(2)][Ce(1)]*
(24)
-i302)(ovd)
2
Table 1
The Gibbs energy (g = h - Ts) parameters used to construct the theoretical g(OJ, %O,) and S(0,) curves in CeO,_, PuO,_,
in units of J/mol. AGCEOV
CeO,-, PUO, --x
CcO,_, PLlO,_,
PuO,_,
AGCE34(0-1)
-4.6
Ag(O’~(l))
Ag(O*-(2))
Ag(O*-(3))
-4.46x10”-(-1.46)T 2.62x lo3 - 1.07T
4.46 x lo3 - 1.46T - 4.71 X lo3 - ( - 2.67)T
A g(V,(2))
A g(V,(3N
5.35 x lo3 - 1.76T 6.28 x lo3 - 2.13T AsW,(lN
CeO,_,
AGCEB
393.8 x lo3 - 132.6T 451.9~ lo3 - 104.6T
2.51 x lo3 - 0.84T 1.26 x lo3 - 0.40T
0-OT x103-(-1.26)T
-2.51x103-(-0.84)T -8.79x103-(-2.78)T
13.05x lo3 -3.47T 20.92x lo3 - 6.28T
50.21 X lo3 - 16.747 56.48 x lo3 - 11.92T
AGCE34(0-2) -7.32x103-(-1.88)T -10.46x lo3 -(-5.02)T Ag(O*-(4)) -4.71X103-(-2.67)T -9.41X103-(--3.20)T A sW,(4N
- 62.76 x lo3 -( - 25.10)T -69.04x103-(-29.81)T
and
22
,
(4
CeO2-x
This
study
----
-
636OC
: ro8o”c ---____-_,ijooy
900
0 E .
2
j ‘i j
J
700 0
0.10
,
0 20
I
0
0 10
020
030 x
X
Fig. 4. ~ h(Oz) (a) and - .$O,) (b) versus x relationship in CeO,_,. (0) Bevan and Kordis [2], (0) Campserveux and Gerdanian [3], ( A 1 Panlener et al. [21]. For both X(0,) and S(Oz) theoretical curves obtained in this study are drawn at three temperatures.
This --.--
study
:
750°C
Fig. 5. - %Oz) (a) and - S(O?) (b) versus x relationship in PuO,_,. (0) Chereau et al. [4], (0) this study. For both x(02) and S(O,) theoretical curves obtained in this study are drawn at three temperatures.
A. Nakamura /A
-dW(%o) 2
defect-thermodynamic
a[OS)l aN=x
= cAg(02-(i)) i
AGCEOV
In eq. (24) we used the following abbreviations:
2g(Ce3+(0))
= AGCEA+g(V,(O)) = 2g(Ce3+(0))
-t 4g(Ce4+(1))
+g(V,(O))
- 6g( Ce4+(0)), AGCEB =g(Ce4+(2))
(27)
+g(Ce4+(0))
-2g(Ce4+(1)).
23
x and CeO, _ x
eqs. (24)-(26) as r’
(26) AGCEA=
approach to PuO,
(28)
We can also derive the explicit expression of h(O,) directly from eqs. (23)-(28) utilizing the relationship h(0,) = a(g(O,)/T)/$l/T) I X. As is apparent in eq. (24) for g(O,XCe), in these component terms of x(0,), the partial derivative of the concentrations of the various chemical species with respect to l/T at constant x appears. In short, we need to know the only two a[Ce3’(i>]/a(l/T)I X (i = 1, 2), and for these two, though somewhat lengthy and not given here, we can also derive the explicit analytical expressions from eqs. (14) and (1.5). Finally we obtain S(O,) by the relationship S(O,) = (h(O,) - g(O,))/T. For the numerical evaluations of the model, we need to adjust the total of 24 enthalpy (Ah) and entropy (AS) parameters (12 for each) in 12 Gibbs energy parameters (Ag) listed in table 1. Among these, the only one which is not defined yet by the above equations is AGCEOV. This quantitiy appears in eq. (23) for g(O,) by inserting the three component terms
-g(02-(0))
+ 4g(Ce4+(1))
- 6g(Ce4+(0))
-s(0’-(0)).
(29)
Eq. (29) means that AGCEOV is the Gibbs energy change of the fluorite crystal lattice due to the following defect formation reaction: 6Ce4+(0) + 02-(0) + 2Ce3+(0) + 4Ce4+(1) + V,(O) + O,(g)/2. Then, inspection of the Gibbs energy parameters in table 1 shows that we are simulating how and to what extent the actual defect formation process deviates from the simplest eq. (29) type to the more complex one involving more various kinds of defect species with x and T depending on the relative stability relationship (Ag) of these various defect species (and their various combinations) inside and between Ce and 0 sites.
3. Results and discussions All the Gibbs energy (g), enthalpy (h) and entropy (s) parameters we finally adopted are listed in table 1 for both CeO,_, and PuO,_,. The theoretical z(O,) and $0,) curves for CeO,_, and PuO,_, are shown in figs. 4 and 5, respectively. Those of g(O,)(= RTlnP& here PO*, in unit of MPa divided by the standard state oxygen pressure, 0.1013 MPa) for both
1
Pu o,_, 750°c
9oo'c
11oo'c
f5OO'C
16
-20
-10 Log
PO”
-20 (bl
Log
PO:
Fig. 6. logx versus logP& relationship in CeO,_, (a) and PuO,_, (b). ( -_) This study. (a) (0) Bevan and Kordis [2], (A ) Panlener et al. 1211,(0) Campserveux and Gerdanian [91. (b) (0) Markin and Rand 1161,(0) Swanson [22], (a) Atlas and Schlehman [231,(01 Serensen [24], ( v 1 Woodly [25].
A. Nakamura
24
/A
defect-thermodynamic
systems are shown in fig. 6 as logx versus logI’,:_ curves. Agreement with the reported experimental data in both systems appears to be satisfactory over the wide x and T range for all of these data except for x > 0.26 where experimental -h(O,) and -S(02) both seem to drop more sharply with X. Yet, the most prominent feature of sigmoidal h(0,) shape in CeO, ( is well reproduced by the present model (fig. 4a). For PuO,_, experimental data are rather sparse and their scatters are also relatively large. Chereua et al. [4] evaluated S(0,) of this system by the combination of their own x(0,) data (o in fig. Sa) with g(O:) data of ref. [16] (o in fig. 6b). As the latters seem to give somewhat higher g(O?), we reevaluated S(O,) using the average g(O?) data at 1100°C in fig. 6b. which are shown in fig. .5b as filled circles. The present thoretical curves closely follow these reevaluated S(O,) data. As seen in fig. 6, the present theoretical &O,) curves also reproduce the reported miscibility gap at low temperature and small x region in both systems. According to the present model, the sigmoidal shape of h(Oz) in CeO,_ r is brought about by the coupled complex local configurational changes between Cc3 +,4i on Ce sites and O,y- , V,, on 0 sites with x and T.
z
approach to PuO,
,
Therefore. such sigmoidal shape also appears in S(0,) (fig. 4b), but to the lesser extent due to the existence of the prevalent configurational entropy term itself which steadily shifts S(0,) to the lower value side. We can understand the present situation more clearly in fig. 7. which shows how the three component terms cooperate to produce such a coupled unique shape of both h(O,) and S(0,). Although the major part of the initial drop of --&(02) is due to the OBci term, its sigmoidal shape is mostly due to the internally coupled Ce and OV, terms, the relative contribution of the former Cc term being much greater in its final sharp drop beyond the peak around x = 0.24. The essential “structure” of the Gibbs energy paramcters in table 1 which causes such @O,) and S(OZ) shape and thereupon derived defect structure of these systems arc brictly summarized as follows: (1) very large positive and negative values of h and s of Ag(V(3)) and Ag(V,(4)) and also of themselves, rcspectively. These means that V,(3) and V,(4) are the least and the most favorable state for V,, respectively. For example, {Ag(V,(3)) + Ag(O’- (i))) - (Ag(V,(i) t Ag(O’-(3))} = (g(VJ3)) + g(O*-(iI) - (g(V,,(i)) + R (0’ (3))) have large positive values for any i(=
(
E \ 2
i --__ 0 \ -----..__.____ ----------.. _!6_____ .-.._ ___________ . . . . __ ‘. ‘.__
\
0” lJ= 4
, and CEO,
\
-1 O(
-___
ovd __________----------__I
I
-2013
I 636’C
-301
,
/ O'-
(4 Fig. 7. Relative
111
03
02
0.1 X
variations
of - L(02)
b)
(a) and
- .X0,)
(b) and their Y = 0.001.
component
terms
with
x. References
point:
(a) .r = 0. (h)
A. Nakamura 0, 1, 2, 4),
/ A defect-thermodynamic
which means that the defect equilibria, V,(i) + G:-(3) ti V,(3) + O;-(i), are shifted extremely to the left-hand side for any i avoiding always the formation of ~~(3). If their absolute values, I Ag(V,(3)) I and / Ag(V0(4))l are made smaller, the sigmoidal shape becomes weaker, which is seen in figs. 4 and 5 through their temperature dependence. (2) The combination of AGCE34(0-1) (> 0) (eq. (16)) and AGCE34(0-2) (< 0) (eq. (17)) so as to make the positive value of (2AGCE34(0-1) - AGCE34(0-2)). These means that Ce3+ favors Ce(2) the best, Ce(0) the next and Ce(1) the least, and also the defect equifibria, Ce3’(0) + Ce3+(2) + 2Ce4+(1) * Ce4+(0) + Ce4+(2) f 2Ce3+(l), is shifted to the left-hand side to some extent. The subtlety of these Gibbs energy parameter changes is illustrated by the considerably different shape of x(0,) between CeO,_, and PuO,_, in spite of the fact that the both systems share the common basic “structure” of the Gibbs energy parameters (1) and (2) as seen in table 1. Of course, due to the idealized assumptions adopted in fromulating the model, we need to receive the numerical details of these defects’ Gibbs energy parameters and therefrom derived defect structure of these systems with a grain of salt. As for the assumption (2) shown in fig. 3, the author is now attempting to reformulate the model on the more realistic disordered type treatment 1111, and this situation will become much clearer after some results are obtained from such an attempt. Yet, it is the author’s view that the basic idea of the present approach based on the local defect-defect interaction model and therefrom derived basic interpretation on the coupled strong sigmoidal variation of %(O,) and $0,) is a reasonable one. This view seems to be partly supported by the recent Monte Carlo type simulational results on CeO,_, reported by Boureau et al. 1141: Their results show that the long-range Coulomb interaction between the charged defect causes the decrease of %O,) in the relatively small x region (n < 0.15), but this decrease is approximately + (or less) of the experimental one shown in fig. 4a. For the deeper understanding of the defect structure and the thermodynamic properties of these highly defective actinide and related fluorite-type oxides involving both the local (short-range) and long-range interaction effects, much remains to be done in the future through the mutual interaction and cooperation between the Present type more phenomenological approach and the more fundamental defect-physical approach such as the above Monte Carlo [12-141 and the lattice-dynamical simulation techniques [26].
approach to FuOz _ x and CeO, _ x
25
4. Conclusion In this paper, the author proposed a defect-thermodynamic approach to CeO,_, and PuO,_, based on the first N.N. Ce3+*4+-V,(02-) interaction model. Under some simplified assumptions on V, arrangement on the oxygen sublattice, the number of local configurations (the chemical species) is reduced to 16, all their concentrations given by simple equations. Given their Gibbs energy parameters (g), we can derive the relatively simple analytical expression of g(O,), h(0,) and $0,) which are solved numerically on computer. With the proper choice of the g parameters we can reproduce well all the thermod~amic data of oxygen of these systems over the wide x and T range. The g parameters indicate that Ce3+(2), Ce3’(0) and Ce3’(1) are the most, the next and the least stable local states for Ce3’ respectively, and V,(3) and V,(4) are the urmost unstable and the urmost stable local states for V,, respectively. The onset of the sigmoidal x(0,) and $0,) in these systems is interpreted as a result of the coupled local configurational changes between Ce and 0 sites with x and T. Irrespective of the idealized and phenomenological nature of the model, the author believes that the basic idea of the present approach based on the local defect-defect interaction model is a reasonable one and combined with other various approach paves the way to our deeper understanding on the thermodynamic and defect properties of these actinide and related ~uorite-type oxides.
References [ll A. Nakamura and T. Fujino, J. Nucl. Mater. 167 (1989) 121 “,4;. Bevan and J. Kordis, J. Inorg. Nucl. Chem 26 (1964) 1509. 131J. Campserveux and P. Gerdanian, J. Chem. Thermodyn. 6 (1974) 795. 141 P. Chereau, G. Dean, M. De France and P. Gerdanian, J. Chem. Thermodyn. 9 (1977) 211. [5j A. Nakamura and T. Fujino, J. Nucl. Mater. 149 (1987) 80. [61 T.B. Lindemer, Calphad 10 (1986) 129. f71 M. Hillert and BO Jansson, J. Am. Ceram. Sot. 69 (1986) 732. [8l L.M. Atlas, J. Phys. Chem. SoIids 29 (1968) 91. 191J. Campserveux and P. Gerdanian, J. Solid State Chem. 23 (1978) 73. [lOI L. Manes, Nonstoichiometric Oxides, ed. T. Sorensen (Academic Press, New York, 1981) p.p. 99-154. 1111G. Boureau and J. Campserveux, Philos. Mag. 36 (1977) 9.
26
A. Nakamura
/A
defect-thermodynamic
1121 R. Tetot, M. Benzakour and G. Boureau. .I. Phys. Chem. Solids 48 (1988) 381. [13] G. Boureau and R. Tetot, Phys. Rev. B40 (1989) 2304 and 2311. [I41 R. Tetot, M. Benzakour and G. Boureau, J. Phys. Chem. Solids 51 (1990) 545. [15] B. Touzelin, J. Nucl. Mater. 101 (1981) 92. [16] T.L. Markin and M.H. Rand, Thermodynamics, vol. I (IAEA, Vienna, 1966) p. 145. [17] J.C. Boivineau, J. Nucl. Mater 60 (lY76) 31. [18] D.J. Kim, J. Am. Ceram. Sot. 72 (1989) 1415. [19] B.G. Hyde and L. Eyring, Rare Earth Research III. ed. L. Eyring (Gordon and Breach, New York, 1965) p. 623. [20] M.R. Thornber and D.J.M. Bevan, Acta Crystallogr. 824 (1968) 1183.
approach to PuO,
* and CeO,
,
[21] R.J. Panlener, R.N. Blumenthal and J.E. Carnier. J. Phys. Chem. Solids 36 (1975) 1213. [22] G.C. Swanson, Los Alamos Scientific Laboratory Report LA-6063-T ( 1975). [23] L.M. Atlas and G.J. Schlehman. Thermodynamics I1 (IAEA, Vienna, 1966) p. 407. [24] O.T. Sgrensen, Plutonium lY75 and Other Actinidcs, eds. J. Blank and R. Lindner (North-Holland, Amsterdam, 1976) p. 123. [2S] R.E. Woodly, J. Nucl. Mater. Y6 (1981) 5. [26] R.W. Grimes, C.R.A. Catlow and A.M. Stoneham. J. Chem. Sot. Trans II, 85 (1989) 335.