.. _section-A9.3:

Appendix 9.3: Thermodynamics and Kinetics Parameters of Fuel Clad Chemical Interaction Model
============================================================================================


*Lanthanide Migration*

The solubility limit of lanthanides in metallic fuel is assumed to be zero [9‑4].
The lanthanide diffusion coefficient is given in :eq:`eq142`.
Below 800 K, the temperature is set to 800 K in order to account for the irradiation-induced diffusivity.
Ref. [9‑5] is used to assess the solubility limit of lanthanides.
Thermodynamic and kinetics parameters related the lanthanide diffusion and precipitation model are given in :numref:`table-A9.3-1`.
Note that solubility limit of D9 cladding is reduced significantly compared to the HT9 cladding in order to account for excessive clad wastage behavior observed in experiments.
Diffusion constant and diffusion activation energy is used as fitting factors.


.. math::
   :label: eq141

    D_{la} = \begin{cases}
    D_0 exp\left( - \frac{Q_0}{RT} \right)  & T \ge 800 \text{ K} \\
    D_0 exp\left( - \frac{Q_0}{800R} \right)  & T \leq 800 \text{ K} \\
	\end{cases}

.. _table-A9.3-1:

.. list-table:: Diffusion coefficient, heat of transport, and Lanthanide yield for Lanthanide migration
    :header-rows: 0
    :align: center
    :widths: auto

    * - :math:`D_{0}` (m :math:`^2`/s)
      - 5.0
    * - :math:`Q_{0}` (kJ/mol)
      - 250.0
    * - :math:`Y_{la}`  (#/J)
      - :math:`15.29E+09`
    * - Solubility in HT9 (atom/ :math:`m^{3}`)
      - :math:`6.4E+27`
    * - Solubility in D9  (atom/ :math:`m^{3}`)
      - :math:`0.5E+27`


*Iron and Actinide Migration*

The iron and actinide diffusion coefficients are calculated using the
relationships given in :eq:`eq142` and :eq:`eq143`, respectively. :numref:`table-A9.3-2` and :numref:`table-A9.3-3`
gives the diffusion constant and activation energy
values. :numref:`table-A9.3-4a` and :numref:`table-A9.3-4b` gives the thermodynamics and kinetics parameters used
in iron and actinide diffusion and eutectic formation. The experimental
observartions in Ref. [9‑6] is used to extract thermodynamics and
kinetics parameters.

.. math::
   :label: eq142

   D_{Fe} = D_{0Fe} exp \left( - \frac{Q_{0Fe}}{RT} \right)

.. math::
   :label: eq143

    D_{Ac} = D_{0Ac} exp \left( - \frac{Q_{0Ac}}{RT} \right)

.. _table-A9.3-2:

.. list-table:: Diffusion constant and activation energy for iron diffusion in metal fuel
    :header-rows: 1
    :align: center
    :widths: auto

    * -  Parameter
      -  T < :math:`T_{\beta + \gamma }`
      -  T :math:`\geq T_{\beta + \gamma }`
    * -  :math:`D_{0Fe}` (m :math:`^2`/s)
      -  0.225E-03
      -  1.0E-03
    * -  :math:`Q_{0Fe}` (kJ/mol)
      -  220
      -  220

.. _table-A9.3-2b:

.. list-table:: Diffusion constant and activation energy for iron diffusion in cladding
    :header-rows: 0
    :align: center
    :widths: auto

    * -  :math:`D_{0Fe}` (m :math:`^2`/s)
      -  8.5E-03
    * -  :math:`Q_{0Fe}` (kJ/mol)
      -  251

.. _table-A9.3-3:

.. list-table:: Diffusion constant and activation energy for Actinides and eutectic
    :header-rows: 1
    :align: center
    :widths: auto

    * -  Parameter
      -  T < 1074.15 K
      -  1074.15 K :math:`\leq` T :math:`\leq` 1094.15 K
      -  T :math:`\geq` 1094.15 K
    * -  :math:`D_{0Ac}` (m :math:`^2`/s)
      -  2.86E-08
      -  :math:`2.86E-08 + 4.29E-08 \frac{T-1075.15}{20}`
      -  7.15E-08
    * -  :math:`Q_{0Ac}` (kJ/mol)
      -  120
      -  120
      -  120

.. _table-A9.3-4a:

.. list-table:: Thermodynamics and kinetics parameters for iron, actinide diffusion, and eutectic formation
    :header-rows: 1
    :align: center
    :widths: auto

    * - Phase Structure
      - Interaction Parameter

        (J/mol)
      - Iron Solubility

        Limit (%)
      - Phase growth activation

        energy (kJ/mol)
    * - (U,Pu)Fe2
      - -3000
      - 66.7
      - 220
    * - Fe2Zr + (U,Pu)6Fe
      - -1200
      - 40.5
      - 220
    * - (U,Pu)6Fe + U-32Zr-50Fe
      - -750
      - 28.6
      - 220
    * - (U,Pu)6Fe + U-42Zr-33Fe
      - -650
      - 21.8
      - 220
    * - U-Zr matrix + U-42Zr-33Fe
      - -500
      - 13.2
      - 220
    * - U-Zr matrix + U-23Zr-6Fe
      - -800
      - 2.5
      - 220
    * - (U,Pu)Zr
      - -800
      - 0.0
      - 220
    * - Clad
      - -600
      - 0.05
      - 220
    * - Eutectic
      - -1600
      - 0.5
      - 220

.. _table-A9.3-4b:

.. list-table:: Phase growth constansts (1/s) for iron, actinide diffusion, and eutectic formation
    :header-rows: 1
    :align: center
    :widths: auto

    * - Phase Structure
      - T  :math:`\le T_{\beta + \gamma}`
      - T  :math:`\geq T_{\beta + \gamma}`
    * - (U,Pu)Fe2
      - 1.69E+05
      - 7.5E+05
    * - Fe2Zr + (U,Pu)6Fe
      - 3.38E+05
      - 1.5E+56
    * - (U,Pu)6Fe + U-32Zr-50Fe
      - 1.7E+06
      - 7.5E+06
    * - (U,Pu)6Fe + U-42Zr-33Fe
      - 2.81E+06
      - 1.25E+07
    * - U-Zr matrix + U-42Zr-33Fe
      - 3.94E+06
      - 1.75E+07
    * - U-Zr matrix + U-23Zr-6Fe
      - 5.63E+07
      - 1.5E+08
    * - (U,Pu)Zr
      - 0.0
      - 0.0
    * - Clad
      - 0.0
      - 0.0
    * - Eutectic
      - 2.81E+09
      - 1.25E+10