## 2 Background for Modelling Gas Dissolution into and through Metals  
Figure 1 displays the expected process undergone by a diatomic gas when passing through a generic metal barrier. In this process, generally the gas phase dynamics (i) evolve much faster than the diffusion (iii) in the solid medium [9]. Thus:  
* the gas can be always considered in equilibrium for the purpose of evaluating thermodynamic properties (pressure, density,...) that affect the gas-metal interaction; and
* the diffusion is mainly determined by surface dynamics, and not gas availability outside of the metal. For any given gas-metal pair, the surface dynamics (ii and iv) characterize the process as either surface-controlled or bulk-controlled. In surface-controlled systems the gas intake rate through the interface limits the diffusion process, due to slow adsorption or chemical reactions. In bulk-controlled systems, the gas dissolution into the metal is only determined by the medium's diffusivity, because the concentration at the interface is high enough to supply the fluxes determined by the diffusive process [10]. It is generally agreed that hydrogen isotopes diffusion into Pb-Li and steel are bulk-controlled systems [11; 12; 13; 14; 10], although exceptions have been reported for Pb-Li [9]. Figure 1: Diagram of the passage of a diatomic gas (\(X_{2}\)) through a generic metal barrier. The process is subdivided in four steps. (i) Free gas expansion, to reach the gas-metal interface (i.e., the metal surface). (ii) Adsorption + Dissociation, at the interface. (iii) Diffusion, through the metallic medium in the direction of lower concentrations. (iv) Recombination + Desorption, at the other gas-metal interface. The diffusion process (iii) has two distinct stages: a diffusive one, that is transient and lasts until medium saturation, and a permeative one, when in steady state--a system's steady-state flux is mathematically described by Richardson's Permeation Law [14]. For a metal M and a diatomic gas X\({}_{2}\) in low pressure regimes--a gas is considered in a low pressure regime, and thus the ideal gas law may be applied, when its reduced pressure is much smaller than 1, i.e., its pressure is far from its critical point [15; 16]--both stages are governed by the diffusivity \(\mathcal{D}_{X}^{M}\) (m\({}^{2}\)/s) and the Sieverts' constant \(\mathcal{S}_{X_{2}}^{M}\) (mol/m\({}^{3}\)/\(\overline{\
u}\)B). This process can be mathematically described by Fick's Law--Equation (1)--and the principle of mass continuity--Equation (2)--which respectively relate: the gradient of the local concentration of a species to the molar flux of this species, and the divergence of such flux to the time variation of that concentration. Saturation profiles depend on the boundary conditions, which can be determined using Sieverts' Law--Equation (3). \[\overrightarrow{Q_{X}}=-\mathcal{D}_{X}^{M}\cdot\overrightarrow{ \
abla}\,c_{X} \tag{1}\] \[\frac{\partial}{\partial t}\,c_{X}+\mathbf{\
abla}\,\mathbf{\cdot}\,( \overrightarrow{Q_{X}})=0\] (2) \[c_{X}^{*}=\mathcal{S}_{X_{2}}^{M}\cdot\sqrt{p_{X_{2}}} \tag{3}\]  
In the equations, \(c_{X}\) (mol/m\({}^{3}\)) is the metal's concentration of atoms \(X\) and \(c_{X}^{*}\) is the Sieverts' equilibrium concentration; \(\overrightarrow{Q_{X}}\) (mol/m\({}^{2}\)s) is the molar flux of \(X\); \(p_{X_{2}}\) (Pa) is the partial pressure of \(X_{2}\) in the gas phase [14]. Dimensional analysis shows that the diffusivity displays physical unit of time, while the Sieverts' constant does not. This means a varying experimental quantity that represents the process, such as the pressure, can be used to measure:  
* diffusivity, by verifying time-related properties of a pressure curve characterization;
* Sieverts' constant, by comparing initial and final states of a pressure curve characterization;  
in a given static experiment [14]. Both \(\mathcal{D}_{X}^{M}\) and \(\mathcal{S}_{X_{2}}^{M}\) display an Arrhenius dependency to temperature \(T\) (\(K\)), as shown in Equations (4) and (5).