As noted in Chapter 2.01, The Actinides Elements: Properties and Characteristics; Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides; and Chapter 2.20, Fission Product Chemistry in Oxide Fuels, modeling of complex systems such as U-Pu-Zr and (U, Pu)O2±x has been exceptionally difficult. For example, actinide oxide fuel is understood to be nonstoi­chiometric almost exclusively due to oxygen site vacancies and interstitials. As a result, the fluorite — structure phase has been treated as being composed of various metal-oxygen species with no vacancies on the metal lattice.

An early and successful modeling approach has employed a largely empirical use of variable stoichi­ometry species that are mixed as subregular solutions to fit experimental information.17-20 This technique can be viewed as a variant of the associate species method.21 In the approach, thermochemical values were determined from the phase equilibria, that is, the phase boundaries, and data for the temperature — composition-oxygen potential [mO2 = RT ln(pO*)] where pO2 is a dimensionless quantity defined by the oxygen pressure divided by the standard state pressure of 1 bar. The UO2±x phase, for example, was treated as a solid solution of UO2 and UaOb where the values of a and b were determined by a fit to experimental data. Figure 3 illustrates the trial and error process using a limited data set to obtain the species stoichiometry which results in the best fit to the data. As can be seen, a variety of stoichio­metries for the constituent species yield differing curves ofln(pO*) versus fx), with the most appropri­ate matching the slope of 2. Thus, for this example U10/3O23/3 provides for an optimum fit between U3O7 and U4O9, and its solution with UO2 best reproduces the observed oxygen potential behavior. Utilizing a much more extensive data set from a variety of sources resulted in a set of best fits to the data, yet they required three solid solutions to ade­quately represent the entire compositional range for UO2±x These are

-4

-24

Figure 3 The іп(рО|) dependence as a function of x for UO2+x and of f(x) for several solid-solution species’ stoichiometries for an illustrative oxygen pressure — temperature-composition data set. Coincidence with the theoretical slope of 2 indicates the proper solution model. Reproduced from Lindemer, T. B.; Besmann, T. M. J. Nucl. Mater. 1985, 130, 473-488.

UO2+x (high hyperstoichiometry, i. e., large values of x): UO2 + U3O7,

UO2+x (low hyperstoichiometry, i. e., smaller

values of x): UO2 + U2O45, and

UO2-x (hypostoichiometric): UO2 + U1/3.

The results of the models for UO2±x are plotted in Figure 4 together with the entire data set used for optimizing the system.

The above models for UO2±x have been widely adopted, as has been a similar model of PuO2-x.16 These have also been combined to construct a suc­cessful model for (U, Pu)O2±x.16 Lewis eta/.22 used an analogous technique for UO2±x. Lindemer23 and Runevall et a/.24 have generated successful models of CeO2-x. Runevall et a/.24 also used the method for NpO2-x, AmO2-x (with the work of Thiriet and Konings25), (U, Am)O2±x, (Th, U)O2±x, (U, Ce)O2_* (Pu, Am)O2±x, and (U, Pu, Am)O2±x. They noted that results for the (Th, U)O2+x were less successful per­haps because of the difficulty in the measurements

U-O liquid region
0
-200
10-3
-600
X =0.3
U-O liquid region

1000 1500 2000 2500 3000

Temperature (K)

Figure 4 Oxygen potential plotted versus x for the models of UO2±x of Lindemer and Besmann17 overlaid with the entire data set used for the optimization.

made near stoichiometry. Osaka et a/.26-28 used the approach to successfully represent the (U, Am)O2_x, (U, Pu, Am)O2_x and (Am, Th)O2_x phases.