## 3 A Methodology to Differentiate Dissolution from Gas Losses in Gas-Metal Systems
### Boundary Conditions (BC) on a Double Metallic Barrier for the Dissolution of a Diatomic Gas  
The boundary conditions (BCs) on the surface of a metal for a dissolution process simulated using this approach depend on the type of system being simulated. Bulk-controlled systems lead to Dirichlet BCs, i.e., the concentration value--in this work it is assumed that the gas phase density and the gas concentration in the solid can be used interchangeably--at the gas-metal interface is directly applied as BC, since dissociation and recombination at the surface are not limiting the diffusion process. On the other hand, surface-controlled systems lead to Neumann BCs, i.e., the derivative of the concentration at the gas-metal interface must be applied as BC, since surface fluxes limit the diffusion process. To apply this model to steel and Pb-Li, it is assumed that these metals are bulk-controlled (_vide_ Section 2) and thus require only Dirichlet BCs; the methodology can nevertheless be extended to surface-controlled systems if surface dynamics coefficients are known (e.g., [13]). As displayed in Figure 3, three possible types of BC are foreseen in the proposed model:  
* **Coupled BC**: gas-metal interfaces exposed to a finite amount of gas experience a BC that changes in time, since the flow of particles through the metal surface alters the gas density outside the metal. If more than one diffusion channel is simulated, all BCs must be altered in unison, taking into account the amount of gas flowing through the boundary node of each channel. For this reason, these BCs are effectively coupled. * Co-Dependent BC: metal-metal interfaces only present a change in medium properties and no other process is assumed (e.g., recombination/dissociation); that is, fluxes are not determined by surface-controlled aspects. * Fixed BC: gas-metal interfaces exposed to a gas reservoir of infinite supply experience a BC constant in time. In the particular case that the BC is equal to zero (as shown in the example in Figure 3), the model represents a surface that acts as a perfect sink (i.e., that metallic surface is constantly and efficiently ventilated). Figure 4 illustrates, through the simplified example of a single-metal barrier under equal conditions on both sides, how Fixed BCs are determined. The ideal gas law is used to compute a molecular density \(c_{X_{2}}\) outside of the metal using the gas pressure (i). Due to surface dissociation, the interface experiences an atomic density \(c_{X}\) (ii). Since Sieverts' Law establishes the atomic concentration of gas present inside the metal when in equilibrium (given that gas phase pressure \(p_{X_{2}}\)), Sieverts' concentration \(c_{X}^{*}\) is taken as a critical limit (maximum) to this concentration (iii). This agrees with the expected behavior of the diffusion process, that tends to a constant profile equal to \(c_{X}^{*}\) when in steady-state. Thus, the BC is taken as the minimum between \(c_{X}^{*}\) and the gas atomic density at the interface \(c_{X}\)  
The procedure used for determining Fixed BCs can be applied to cases in which the gas pressure \(p_{X_{2}}\) varies in time. In these cases, since both \(c_{X}\) and \(c_{X}^{*}\) change in time, tasks (i)-(iii) are performed at every time step. In the particular case in which the fluxes through the gas-metal interface alter the pressure and multiple channels are simulated, the model presents Coupled BCs. One example of such scenarios is the simulation of a closed metal container during a static absorption experiment. A simulation with the model presented in this work can be used to compute the variation of the inner pressure in the container, caused by gas loss. Each container wall can be modelled by one diffusion channel, and the total gas loss in the container at each time step can be computed by multiplying the average atomic fluxes through each wall by that wall's area.