## 2 Background for Modelling Gas Dissolution into and through Metals
\[\mathcal{D}=D_{\infty}\,\cdot\,e^{-\frac{E_{d}}{K\cdot T}} \tag{4}\] \[\mathcal{S}=S_{\infty}\,\cdot\,e^{-\frac{E_{s}}{K\cdot T}} \tag{5}\]  
The terms \(E_{d}\) and \(E_{s}\) are referred to as _activation energies_ (J/mol). For the purposes of this work, the terms \(D_{\infty}\) and \(S_{\infty}\) will be called _infinite-temperature_ coefficients. All four terms (\(D_{\infty}\), \(S_{\infty}\), \(E_{d}\) and \(E_{s}\)) will be addressed as _dissolution parameters_. For a given set of temperature-dependent measurements, the dissolution parameters can be obtained by linear regression of data plotted, in logarithmic scale, as function of the reciprocal of the temperature (1/\(\tau\)) [14]. Figure 2 shows a literature review of Arrhenius functions of diffusivity and Sieverts' constant for the dissolution of different hydrogen isotopes (Q) in Pb-Li. Notice the scattering over orders of magnitude, especially in Figure 2b. When only point measurements were available, the curves were interpolated (but not extrapolated) using logarithmic regression. Sieverts' constants are sometimes presented in literature in different units (e.g., the concentration represented in atomic fraction, i.e., \([\mathcal{S}_{X_{2}}^{M}]=\
icefrac{{1}}{{\sqrt{p_{\text{a}}}}}\)). In some of these cases the values had to be transformed using the temperature-dependent mass density for Pb-Li [17], which was done _before_ applying the regression. Some references reviewed in Figure 2a,b present not only experimental measurements, but also values obtained from theoretical models (e.g., thermodynamic model for alloys [26; 32; 33]). Some of those apply an extrapolation to obtain coefficients for one isotope based on the coefficient for another. This can be done using Equation (6) due to inertia's impact on the diffusivity. \[\mathcal{D}_{X}^{\mathrm{M}}\propto\left(\frac{m_{M}+m_{X}}{m_{M}\cdot m_{X}} \right)^{\
icefrac{{1}}{{2}}}\implies\frac{\mathcal{D}_{X^{\prime}}^{\mathrm{M} }}{\mathcal{D}_{X}^{\mathrm{M}}}\approx\sqrt{\frac{m_{X}}{m_{X^{\prime}}}} \tag{6}\]  
Figure 2: Literature review of \(\mathcal{D}_{\mathrm{Q}}^{\mathrm{Pb-Li}}\) and \(\mathcal{S}_{\mathrm{Q}}^{\mathrm{Pb-Li}}\). If only point values were available in any given reference, dissolution parameters were obtained by logarithmic regression after applying unit transformation. (**a**) Literature values and functions of diffusivity for hydrogen isotopes in Pb-Li [10; 18; 19; 20; 21; 22; 23; 24]. (**b**) Literature values and functions of Sieverts’ constant for hydrogen isotopes dissolution in Pb-Li [11; 18; 20; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. When necessary, the temperature-dependent density of Pb-Li was used for unit conversion [17]. In it, \(m_{X}\), \(m_{Y^{\prime}}\) and \(m_{M}\) are the atomic masses of two gas isotopes and the metal, respectively [23]. Some references also apply this rationale to Sieverts' constant (e.g., [22]), however it is still debatable whether an isotopic dependence for \(\mathcal{S}_{X_{2}}^{M}\) should be based on zero-point energy analysis instead [26; 33; 36; 37; 38]. A similar literature review of the diffusivity and Sieverts' constant for hydrogen isotopes dissolution in steel was also performed, as it was relevant for the work hereinafter. Graphs analogous to those presented here are shown in Section 4, alongside results of that section.