Phase Relationships in the Carbon–Titanium–Uranium System for Ultra-High Temperature Nuclear Fuels

Abstract

Carbides of uranium have attracted interest as fuels for nuclear thermal propulsion (NTP) to drive deep-space exploration owing to their attractive thermal and neutronic properties. Optimization of NTP technology, however, requires ultra-high temperature reactor environments to maximize the ratio of thrust to propellant to achieve peak rocket engine efficiency. Incorporation of transition metals into uranium carbides offers a pathway to increase the melting point of carbide fuels to address the operational challenges posed by NTP. A thermodynamic model has been developed to examine the phase relationships in the C–Ti–U system at NTP conditions. Calculated phase relationships at ultra-high temperatures predict a stable (U, Ti)C solid solution. Additionally, experimental work on C–Ti–U synthesis via arc melting has been carried out, providing data on phase constitution and compositions that can be used to further refine the thermodynamic model that has been developed.

Appendix

As a supplement to this article, descriptions of the models used to calculate the liquid and solid-solution phases are described below.

Liquid Phase

The Two Sublattice Partially Ionic Liquid (TSPIL) model²⁴ is used to model liquid phases as

$$ \left( {C_{i}^{{ + v_{i} }} } \right)_{P} \left( {A_{j}^{{ - v_{j} }} ,{\text{VaB}}_{{\text{k}}}^{0} } \right)_{{\text{Q}}} $$

(3)

where C, A, VA and B denote cation, anion, vacancy, and neutrally charged species, respectively. Charge neutrality necessitates that Q and P vary such that

$$ P = \sum_{{\text{A}}} \upsilon_{{\text{A}}} y_{{\text{S}}} + Qy_{{{\text{VA}}}} $$

(4)

$$ Q = \sum_{{\text{C}}} v_{{\text{C}}} y_{{\text{C}}} $$

(5)

v_A and y_A are the charge and site fractions of the anion species, and v_C and y_C are the charge and site fraction of the cation species C, respectively. The Gibbs energy of an ionic liquid can be expressed as

$$ \begin{aligned} G_{{\text{m}}} = & \sum \sum y_{{{\text{C}}_{i} }} y_{{{\text{A}}_{i} }}^\circ G_{{{\text{C}}_{i} :{\text{A}}_{i} }} + Q\left( {y_{{{\text{Va}}}} \sum y_{{C_{i} }}^\circ G_{{C_{i} }} + \sum y_{{{\text{B}}_{{\text{k}}} }}^\circ G_{{{\text{B}}_{{\text{k}}} }} } \right) \\ & + {\text{RT}} \left[ {P\sum y_{{{\text{C}}_{i} }} \ln y_{{{\text{C}}_{i} }} + Q\left( {\sum y_{{{\text{A}}_{j } }} \ln y_{{{\text{A}}_{j} }} + y_{{{\text{Va}}}} \ln y_{{{\text{Va}}}} + \sum y_{{{\text{B}}_{k} }} \ln y_{{{\text{B}}_{k} }} } \right)} \right] + {}^{{\text{E}}}G_{{\text{m}}} \\ \end{aligned} $$

(6)

where $^\circ G_{{{\text{C}}_{i} :{\text{A}}_{i} }}$ is the Gibbs energy of formation for v_i + v_j moles of atoms of the endmembers C_iA_j while $^\circ G_{{{\text{C}}_{i} :{\text{A}}_{i} }}$ and $^\circ G_{{{\text{C}}_{i} :{\text{A}}_{i} }}$ are the formation values for C_i and B_k.

Solid Solutions

The solid solutions were modeled using the compound energy formalism (CEF) was introduced by Hillert³⁹ to describe the Gibbs energy of solid phases with sublattices. These phases have two or more sublattices, and at least one of these sublattices has a variable composition. Ideal entropy of mixing is assumed on each sublattice. This model is generally used to model crystalline solids, but it can also be extended to model ionic liquids.

Here, a solution phases with two sublattices, (A, B)a (C, D)b, will be used as an example to illustrate the compound energy formalism. In this model, components A and B can mix randomly on the first sublattice, as do the components C and D on the second sublattice. a and b are the corresponding stoichiometric coefficients. Site fraction $y_{i}^{{\text{s}}}$ is introduced to describe the constitution of the phase and is defined as follows:

$$ y_{i}^{{\text{s}}} = \frac{{n_{i}^{{\text{s}}} }}{{N^{{\text{s}}} }} $$

(7)

$n_{i}^{{\text{s}}}$ is the number of component i on sublattice (s) and N^S is the total number of sites on the same sublattice. When vacancies are considered in the model, the site fraction becomes

$$ y_{i}^{{\text{s}}} = \frac{{n_{i}^{{\text{s}}} }}{{n_{{{\text{VA}}}}^{{\text{s}}} + \mathop \sum \nolimits_{i} n_{i}^{{\text{s}}} }} $$

(8)

$n_{{{\text{VA}}}}^{{\text{s}}} { }$ is the number of vacancies on sublattice (s). The site fraction can be transferred to mole fraction (x_i) using the equation

$$ x_{i} = \frac{{\mathop \sum \nolimits_{{\text{s}}} \,n^{{\text{s}}} y_{i}^{{\text{s}}} }}{{\mathop \sum \nolimits_{i} \,n^{s} \left( {1 - y_{{{\text{VA}}}}^{{\text{s}}} } \right)}} $$

(9)

When each sublattice is only occupied by one component, then end-members of the phase are produced. In the present case, four endmembers exist. They are A_aC_b, A_aD_b, B_aC_b, and B_aD_b. The surface of reference ^refG_m is expressed as

$$ {}^{{{\text{ref}}}}G_{{\text{m }}} = y^{1} y^{2} {}^{^\circ }G_{{\text{A:C}}} + y^{1} y^{2} {}^{^\circ }G_{{\text{A:D}}} + y^{1} y^{2} {}^{^\circ }G_{{\text{B:C}}} + y^{1} y^{2} {}^{^\circ }G_{{\text{B:D}}} $$

(10)

The ideal entropy (^idS_m) and the excess free energy are expressed as

$$ {}^{{{\text{id}}}}S = - R\left[ {a\left( {y_{{\text{A}}}^{1} \ln y_{{\text{A}}}^{1} + y_{{\text{B}}}^{1} \ln y_{{\text{B}}}^{1} } \right) + b\left( {y_{{\text{C}}}^{2} \ln y_{{\text{C}}}^{2} + y_{{\text{D}}}^{2} \ln y_{{\text{D}}}^{2} } \right)} \right] $$

(11)

$$ {}^{{\text{E}}}G_{{\text{m}}} = y_{{\text{A}}}^{1} y_{{\text{B}}}^{1} \left( {y_{{\text{C}}}^{2} L_{{\text{A,B:D}}} + y_{D}^{2} L_{{\text{A,B:D}}} } \right) + y_{{\text{C}}}^{2} y_{{\text{D}}}^{2} \left( {y_{{\text{A}}}^{1} L_{{\text{A:C,D}}} + y_{{\text{B}}}^{1} L_{{\text{B:C,D}}} } \right) $$

(12)

The binary interaction parameters L_i,:k represent the interaction between the constituents i and j in the first sublattice when the second sublattice is only occupied by constituent k. These parameters can be further expanded with Redlich–Kister polynomial as

$$ L_{i,j:k} = \mathop \sum \limits_{\nu } \left( {y_{i}^{1} - y_{j}^{1} } \right)^{\nu } \cdot^{\nu } L_{i,j:k} $$

(13)

In the case of a three sublattice model:

$$ G_{m} = \mathop \sum \limits_{i} y_{i}^{I} \mathop \sum \limits_{j} y_{j}^{II} \mathop \sum \limits_{k} y_{k}^{III} {}^{^\circ }G_{i,j,k} + RT\sum\limits_{s} {\sum\limits_{i} {a^{s} y_{i}^{s} \ln y_{i}^{s} +^{E} G_{m} } } $$

(14)

$$ \begin{aligned} {}^{{\text{E}}}G_{{\text{m}}} = & \mathop \sum \limits_{i} y_{i}^{I} \mathop \sum \limits_{j} y_{j}^{{{\text{II}}}} \mathop \sum \limits_{k} y_{k}^{{{\text{III}}}} \left[ {\mathop \sum \limits_{l > i} y_{l}^{I} \mathop \sum \limits_{\nu } {}_{{}}^{\nu } L_{i,l:j:k} \left( {y_{i}^{I} - y_{l}^{I} } \right)^{\nu } } \right. \\ & + \left[ {\mathop \sum \limits_{l > j} y_{l}^{{{\text{II}}}} \mathop \sum \limits_{\nu }^{v} L_{i:j,l:k} \left( {y_{j}^{{{\text{II}}}} - y_{l}^{{{\text{II}}}} } \right)^{\nu } } \right. + \left[ {\mathop \sum \limits_{l > k} y_{l}^{{{\text{III}}}} \mathop \sum \limits_{\nu } {}_{{}}^{\nu } L_{i:j:k,l} \left( {y_{k}^{{{\text{III}}}} - y_{l}^{{{\text{III}}}} } \right)^{\nu } } \right. \\ \end{aligned} $$

(15)