Intragranular Bubble Behaviour

This model updates the intragranular bubble population and derived quantities such as atoms per bubble, bubble radius, and intragranular gaseous swelling. The bubble concentration is evolved with a simple source–decay law, while the bubble radius is inferred from the total intragranular bubble volume accumulated from gas inventories stored in intragranular bubbles.

Activation and option flag

The model behaviour is selected by the input option:

  • iIntraGranularBubbleBehavior

Option summary

The following cases are implemented:

  • 0: constant bubble concentration and radius (fixed values)

  • 1: Pizzocri et al. (2018) model rates (uses system resolution/nucleation rates)

  • 2: White and Tucker (1983) correlation for initial bubble concentration

  • 3: special case for annealing / helium behaviour using the intragranular similarity ratio

  • 99: no intragranular bubbles (all related variables set to zero)

Inputs

The model uses:

  • iIntraGranularBubbleBehavior (input option)

  • Time step (physics variable)

  • Gas inventories in intragranular bubbles, e.g. Xe in intragranular bubbles, Kr in intragranular bubbles, He in intragranular bubbles (sciantix variables)

  • Intragranular similarity ratio (sciantix variable, used in option 3 and updated at the end)

It also uses system-level quantities from sciantix_system:

  • getResolutionRate()

  • getNucleationRate()

  • gas decay rate and restructured matrix flag (to filter systems)

Model formulation

Bubble concentration evolution

The intragranular bubble concentration \(N\) is evolved as:

\[\frac{dN}{dt} = -k\,N + s\]

where the coefficients \(k\) and \(s\) depend on the selected option (e.g., resolution and nucleation rates for option 1).

In the code, the update is performed using:

  • solver.Decay(N0, k, s, Δt)

so that:

\[N^{n+1} = \mathrm{Decay}\left(N^{n}, k, s, \Delta t\right)\]

Atoms per bubble and bubble volume

For each gas system (e.g., Xe, Kr, He), the model computes the number of atoms per bubble when the system is not radioactive (decay rate = 0) and the matrix is not restructured:

\[n_i = \frac{C_i}{N}\]

where:

  • \(C_i\) is the inventory of gas \(i\) in intragranular bubbles (e.g., Xe in intragranular bubbles),

  • \(N\) is the intragranular bubble concentration.

If \(N = 0\), then \(n_i = 0\).

The intragranular bubble volume is then accumulated by summing the contribution from each gas species:

\[V_b = \sum_i v_i\,n_i\]

where \(v_i\) is the per-atom volume in the lattice returned by system.getVolumeInLattice().

Bubble radius

The intragranular bubble radius is computed from the total bubble volume via a spherical assumption:

\[R_b = 0.620350491 \, V_b^{1/3}\]

(The numeric factor is the implementation constant used in the code.)

Intragranular gaseous swelling

The intragranular gas bubble swelling is computed as the total bubble volume fraction:

\[S_g = \frac{4}{3}\pi N R_b^3\]

In the implementation:

  • \(\frac{4}{3}\pi\) is represented by 4.188790205.

Similarity ratio (helium annealing case)

At the end of the routine, an intragranular similarity ratio is updated from the helium inventory in intragranular bubbles:

\[\mathrm{SR} = \sqrt{\frac{C_{\mathrm{He}}^{n+1}}{C_{\mathrm{He}}^{n}}}\]

If the initial helium inventory in intragranular bubbles is zero, the similarity ratio is set to zero.

Outputs

The model updates (depending on option and available systems):

  • Intragranular bubble concentration

  • Intragranular bubble radius

  • Intragranular <gas> atoms per bubble (for each relevant gas)

  • Intragranular bubble volume

  • Intragranular gas bubble swelling

  • Intragranular similarity ratio

References