Interdiffusion, Intrinsic Diffusion, Atomic Mobility, and Vacancy Wind Effect in γ(bcc) Uranium-Molybdenum Alloy

Abstract

U-Mo alloys are being developed as low enrichment uranium fuels under the Reduced Enrichment for Research and Test Reactor (RERTR) Program. In order to understand the fundamental diffusion behavior of this system, solid-to-solid pure U vs Mo diffusion couples were assembled and annealed at 923 K, 973 K, 1073 K, 1173 K, and 1273 K (650 °C, 700 °C, 800 °C, 900 °C, and 1000 °C) for various times. The interdiffusion microstructures and concentration profiles were examined via scanning electron microscopy and electron probe microanalysis, respectively. As the Mo concentration increased from 2 to 26 at. pct, the interdiffusion coefficient decreased, while the activation energy increased. A Kirkendall marker plane was clearly identified in each diffusion couple and utilized to determine intrinsic diffusion coefficients. Uranium intrinsically diffused 5-10 times faster than Mo. Molar excess Gibbs free energy of U-Mo alloy was applied to calculate the thermodynamic factor using ideal, regular, and subregular solution models. Based on the intrinsic diffusion coefficients and thermodynamic factors, Manning’s formalism was used to calculate the tracer diffusion coefficients, atomic mobilities, and vacancy wind parameters of U and Mo at the marker composition. The tracer diffusion coefficients and atomic mobilities of U were about five times larger than those of Mo, and the vacancy wind effect increased the intrinsic flux of U by approximately 30 pct.

Appendix: Thermodynamic Description of γ(bcc) Phase in U-Mo System

Thermodynamic activity factors of U, $a_{\rm U} ,$ as a function of N _U at a given temperature, were evaluated using the ideal solution, regular solution, and subregular solution models. The $a_{\rm U}$ is related to N _U by the activity coefficient γ_U as[37]:

$$ a_{\rm U} = \gamma_{\rm U} \cdot N_{\rm U} $$

(A1)

In the ideal solution model, the interaction between U and Mo is ignored,[37] and the γ_U is unity. Therefore, $a_{\rm U}$ is equal to N _U.

In both the regular solution and subregular solution models, γ_U is not unity and can be derived from the molar excess Gibbs free energy, $\Updelta G^{E} .$ This is expressed as[37]

$$ \Updelta G^{E} = N_{\rm{Mo}} \Updelta \bar{G}_{\rm{Mo}}^{E} + N_{\rm U} \Updelta \bar{G}_{\rm U}^{E} = \Upomega N_{\rm{Mo}} N_{\rm U} $$

(A2)

where, Ω is a parameter in the regular solution model and is a function of N _Mo or N _U in the subregular solution model. $\Updelta \bar{G}_{{\text{U \;or\; Mo}}}^{E}$ is the partial molar excess Gibbs free energy of U or Mo and is related to its respective γ as[37]

$$ \Updelta \bar{G}_{i}^{E} = RT\ln \left( {\gamma_{i} } \right) = \Upomega N_{j}^{2} \quad \left( {i \ne j} \right) $$

(A3)

where, R is the molar gas constant with a value of 8.31 J/mol K, and T is the temperature in Kelvin.

The molar excess Gibbs free energy, $\Updelta G^{\text{E}} ,$ of U-Mo alloy (γ phase) in the temperature range from 838 K to 1557 K (565 °C to 1284 °C) is given by Parida[27] as

$$ \Updelta G^{E} = N_{\rm U} \left( {1 - N_{\rm U} } \right)\left\{ {\left( {21.533 - \frac{33.680T}{1000}} \right)N_{\rm U} + \left( {\frac{25.698T}{1000} - 2.120} \right)} \right\} $$

(A4)

In the regular solution model, $\Updelta G^{E}$ at a specific composition and temperature is calculated using Eq. [A4]. Then the value of $\Updelta G^{\text{E}}$ is applied to determine Ω through Eq. [A2]. Finally, γ_U is obtained by substituting the value of Ω into Eq. [A3].

The subregular solution model can be more flexible than the regular solution model because it arbitrarily allows Ω to vary with composition as $\Upomega = a + bN_{\rm U}$.[36] Thus, Eq. [A2] can derive to

$$ \Updelta G^{E} = \left( {a + bN_{\rm U} } \right)N_{\rm U} N_{\rm{Mo}} = N_{\rm U} \left( {1 - N_{\rm U} } \right)\left( {bN_{\rm U} + a} \right) $$

(A5)

where a and b can be determined through equating the corresponding terms in Eqs. [A4] and [A5] and can then be utilized to calculate the value of Ω to obtain γ_U. The value of in $a_{\rm U}$ either the regular or subregular solution models is then simply determined by substituting γ_U into Eq. [A1].