Security and Privacy Notice Disclaimer

Surface Tension and Surface Energy of Uranium Dioxide

Surface Tension of Liquid UO₂

In 1987, Hall, Mortimer, and Mortimer¹ reported results of a critical review of available data on the surface tension of liquid UO₂ and on the surface energy of solid UO₂. Because no new data have been reported since this review, the results of this critical review are recommended. The recommended surface tension of liquid UO₂ at the melting point is the average of measurements by Schins,² Christensen,³ and Bates⁴ with a temperature dependence based on an equation derived by Nikolopoulos and Schulz:⁵

where the surface tension,is in J m^-2 and temperature, T, is in K.

Surface Energy of Solid UO₂

From review of the multi-phase equilibrium measurements of the surface energy of UO₂, Hall et al.¹ concluded that from 273 to 3120 K the surface energy in J m^-2 of solid UO_2.00 probably lies between two lines defined as follows

Line 1

and Line 2

with the mean line between these given by

where temperature, T, is in K.

Hall et al.¹ gave the dependence of the solid surface energy on stoichiometry as

where is the surface energy of UO_2±x in J m^-2.

Hall et al.¹ concluded that the effective surface energy for pores in UO₂, _P, is different from . It is given by

Uncertainties

Surface Tension of Liquid UO₂

The standard error in the average of four measurements^2-4 of the surface tension of liquid UO₂ at the melting point is ± 0.085 J m^-2, which is an uncertainty of approximately ± 17%.

Surface Energy of Solid UO₂

Because experimental estimates of the surface energy of solid UO₂ in the temperature range of 1773 to 2173 K from multi-phase equilibration techniques are uncertain up to ± 70% and the sign of the temperature dependence is not unambiguously determined, Hall et al. gave the uncertainty in Eq.(4) as ± 70%. Although the uncertainty in the dependence on stoichiometry is ±15%, the UO_2±x surface energy uncertainty is > ± 70% because of the UO₂ surface energy uncertainty.

The measurements of the surface tension of liquid UO₂ at the melting point are given in Table 1. The value given by Chasanov has no estimate of uncertainty and has not been included in the assessment by Hall et al.¹ Therefore, the recommended value for the surface tension at the melting point, 0.513 ± 0.085 J m^-2, is the average of the surface tensions given in the first four rows of Table 1. Nikolopoulos and Schulz⁵ calculated the surface tension of liquid UO₂ at three temperatures near the melting point using a theory for ionic liquids developed by Furth.⁷ Their calculated values at 3125, 3175 and 3225 K are respectively 0.521, 0.514, and 0.502 J m^-2. These values are consistent with the average experimental value of the measurements by Schins, Bates, and Christensen. The value obtained by Chasanov⁶ is low relative to this calculation. Inclusion of the value, 0.420 J m^-2, given by Chasanov in the average would give 0.494 J m^-2 for the surface tension at the melting point. This value, recommended by Fink, Leibowitz, and Chasanov,⁸ is low relative to the calculation of Nikolopoulos and Schulz.⁵

Nikolopoulos and Schulz⁵ used their calculations to estimate the temperature dependence of the surface tension of liquid UO₂ near the melting point as .

Combining this result and the average experimental value at the melting point, 0.513 ± 0.085 J m^-2, gives the recommended equation for the surface tension of liquid UO₂, Eq,(1). This equation, recommended by Hall et al.,¹ is also recommended in the assessment by Harding et al.⁹

Surface Energy of Solid UO₂

The experimental data have been most recently reviewed by Hall et al.¹ The variations between the published data are much larger than the published error bars. The large variations in the data have been attributed to stoichiometry variations and to errors in the measurements of the angles (contact angle, grain boundary groove angle, and dihedral angle) from which the surface energy is calculated. Hall et al.¹ commented that the error in the dihedral angle dominates the uncertainty.

Surface energies obtained from multi-phase equilibration studies have been reported by Hodkin and Nicholas¹⁰ (Cu on UO₂), by Nikolopoulos, Nazare and Thummler¹¹ (Ni on UO₂), and by Bratton and Beck¹² (Ni on UO₂). Hodkin and Nicholas¹³ used sessile drop measurements of Cu-Th alloys on UO_2±x to study the effect of stoichiometry. Published data from these studies, shown in Figure 1, illustrate the large variation in the available data. Figure 1 includes the two bounding lines and the mean line defined by Hall et al.¹ (Hall line 1, Hall line 2, and Hall Mean), which are given in Eqs. (2-4), as well as estimates at the melting point. Eberhart¹⁴ used the surface tension of liquid UO₂ to estimate the solid surface energy at the melting point as 0.56 ± 0.09 J m^-2. Deshpande, Desai, and Solomon¹⁵ report the surface tension at the melting point as 0.805 ± 0.06 J m^-2 based on an estimate made by Skapski.¹⁶ Hall et al. commented that both estimation methods are more appropriate for metals than for UO₂ and, in the absence of a theoretical critique, used an average of the two values (0.68 ± 0.06 J m^-2) in their assessment. Hertzian indentation studies reported by Matzke et al.^17-19 gave surface energies at room temperature as a function of O:M ratio. Their published values have been included in Figure 1.

Figure 1

Hall et al.¹ have analyzed the various individual parameters that go into the calculations for the surface energies and tried to estimate best values of each. This analysis has the effect of smoothing out each parameter. Figure 2 shows the surface energies published by Hodkin and Nicholas, Nikolopoulos et al., Bratton and Beck, and Matzke et al., and the recalculated values obtained by Hall et al. Note that this re-analysis by Hall et al. has reversed the slope of the data of Nikolopoulos et al. and increased the magnitude of the slope of the data by Hodkin and Nicholas. Hall et al. stated that indentation results tend to be high because there is often plastic deformation rather than the elastic behavior assumed in the model. Indentation measurements on ThO₂ showed that the surface energy was reduced by 35% if the sample had been preheated so that the oxygen becomes mobile.¹⁷ Assuming a similar effect in UO₂ would reduce the surface energy from 1.8 ± 0.3 J m^-2 to 1.2 ± 0.3 J m^-2. In their re-analysis, Hall et al. applied this correction to the indentation data, as shown in Figure 2.

Figure 2

Figure 3

The re-analyzed data given by Hall et al. with estimated error bars are shown in Figure 3. They commented that the mean value they assumed for the surface energy at the melting point could be in considerable error and the room temperature surface energy for stoichiometric UO₂ is very dependent on the assumed 35% correction for relaxation. They stated that their analysis supports the conclusion made by Fink et al.⁸ that, in view of the scatter in the measurements, there is no clear indication of the temperature dependence within the solid phase. However, their analysis indicates that the surface energy of UO_2.00 is likely to lie in a wedge defined by the two lines given in Eq. (2) and Eq.(3) and shown as dashed lines in Figure 3. Since some of the data, both before and after re-analysis, lie outside this wedge, this recommendation has been made with great reserve. The best values would be expected to lie in the band between these two lines. The mean line in this band is given by Eq. (4) and is shown as a solid line in Figure 3.

Hall et al. also assessed the available data on the variation of surface energy with stoichiometry to obtain Eq.(5). For x > 0.05, the dependence is more pronounced than given by Eq. (5).

The ratio of the grain boundary energy to surface energy on the free surface is a function of the grain boundary groove angle only and is therefore better known than the grain boundary energy. Hall et al. define this ratio as:

where is the grain boundary energy and is the surface energy. This relation is assumed by Hall et al. to be correct over the entire temperature range of solid UO₂. Because the error in the surface energy is so large, ± 70%, the uncertainty in the grain boundary energy calculated from this relation is also large.

Hall et al. discussed pore geometry and defined an empirical surface energy of pores,, which they related to the grain boundary surface energy by

Substitution of Eq. (7) into Eq. (8) gives the relation of the surface energy of pores and the surface energy of UO₂ given in Eq.(6).

Further experimental measurements are needed to determine more accurate values of these quantities. To reduce the uncertainty in the surface tension of liquid UO₂, measurements are needed under controlled conditions. To obtain better data for the solid surface energy using the multi-phase equilibration technique, methods must be developed for greater accuracy in the measurements of the angles, which now have errors on the order of 3^o. Surface energy values accurate to ± 10% require that the dihedral angle must be reproducible to 0.05^o.

1. R. O. A. Hall, M. J. Mortimer, and D. A. Mortimer, Surface Energy Measurements on UO₂, - A Critical Review, J. Nucl. Mater. 148, 237-256 (1987); see also A. Critical Review of the Surface Energy of UO₂, J. Less-Common Metals 121, 341-345 (1986);and R. O. A. Hall and M. J. Mortimer, Effect of Changes in Stoichiometry on the Surface Energy of UO₂, J. Nucl. Mater. 137, 77-85 (1985).



2. H. Schins, On the Surface Tension of Liquid UO₂, J. Nucl. Mater. 78, 215-216 (1978).



3. J. A. Christensen, Battelle-Northwest Laboratory Report, Richland, WA, BNWL-SA-588884-A (1966).



4. J. L. Bates, C. E. McNeilly and J. J. Rasmussen, Material Science Research 5, 11 (1971); see also Battelle-Northwest Laboratory Report BNWL-SA-3579 (1970).



5. O. Nikolopoulos and B. Schulz, Density, Thermal Expansion of Stainless Steel and Interfacial Properties of UO₂ - Stainless Steel Above 1690 K, J. Nucl. Mater. 82, 172-178 (1979).



6. M. G. Chasanov, L. Leibowitz, and S. D. Gabelnick, J. Nucl. Mater. 49, 129 (1974).



7. R. Furth, Proc. Cambr. Phil. Soc. 37, 252 (1941) as referenced by O. Nikolopoulos and B. Schulz, J. Nucl. Mater. 82, 172-178 (1979).



8. J. K. Fink, M. G. Chasanov, and L. Leibowitz, Thermodynamic Properties of Uranium Dioxide, Argonne National Laboratory Report ANL-CEN-RSD-80-3 (1981).



9. J. H. Harding, D. G. Martin, and P. E. Potter, Thermophysical and Thermochemical Properties of Fast Reactor Materials, Commission of the European Communities Report EUR 12402 EN (1989).



10. E. N. Hodkin and M. G. Nicholas, The Surface and Interfacial Energies of Stoichiometric Uranium Dioxide, J. Nucl. Mater. 47, 23 (1973).



11. P. Nikolopoulos, S. Nazare, and F. Thummler, Surface Grain Boundary and Interfacial Energies in UO₂ and UO₂-Ni, J. Nucl. Mater. 71, 89-94 (1977); see also Comments on "Surface Grain Boundary and Interfacial Energies in UO₂ and UO₂-Ni," J. Nucl. Mater. 78, 213-214 (1978).



12. R. J. Bratton and C. W. Beck, Surface Energy of Uranium Dioxide, J. Am. Ceram. Soc. 54, 379-381 (1971).



13. E. N. Hodkin and M. G. Nicholas, Surface and Interfacial Properties of Non-Stoichiometric Uranium Dioxide, J. Nucl. Mater. 67, 171-180 (1977); see also Comments on " Surface Grain Boundary and Interfacial Energies in UO₂ and UO₂-Ni" by P. Nikolopoulos et al., J. Nucl. Mater. 74, 178 (1978);



14. J. G. Eberhart, J. Nucl. Mater. 25, 103 (1968).



15. M. S. Deshpande, P. D. Desai, and a. A. Solomon, High Temp. Sci. 17 303 (1984).



16. A. S. Skapski, Acta Metall. 4 576 (1956), as referenced by R. O. A. Hall, M. J. Mortimer, and D. A. Mortimer, J. Nucl. Mater. 148, 237-256 (1987).



17. H. Matzke, T. Inoue, and R. Warren, The Surface Energy of UO₂ as Determined by Hertzian Indentation, J. Nucl. Mater. 91, 205-220 (1980).



18. H. Matzke, J. Mater. Sci. 15, 739 (1980).



19. H. Matzke, The Fracture Surface energy of (U_0.8 Pu_0.2)O₂, J. Nucl. Mater. 113, 273-275 (1983).