## 3 Methods
### Heat Transport Related to Hydrate Phase Transitions
This results in the inhibition of long-range modes and spatial localization of energy, suggesting that their low thermal conductivity is not only due to guest-host interactions. The rigidity of the framework and hydrate lattice structures will also contribute lower thermal conductivity of clathrates compared to ice. Both available models and experimental data point to a typical hydrate thermal conductivity of 0.45-0.7 W/(m\(\cdot\)K) within the temperature range 265 K to 280 K. Taking methane hydrate as an example, thermal conductivity measured in compacted samples using a needle probe at conditions similar to those in natural environments amounted to 0.62 \(\pm\) 0.02 W/(m\(\cdot\)K) [33]. The only reported value for the CO\({}_{2}\) hydrate thermal conductivity is 0.49 W/(m\(\cdot\)K) [34]. In view of the above, Equation (8) has been reformulated in terms of the mass transport across the hydrate/liquid water interface during the hydrate formation:  
\[ \begin{split} A_{\perp}\left(\frac{dI_{formation}}{dR}\right)\Delta H ^{formation}(T,P)=K\Delta T_{w}+K\Delta T_{H}+\\ A_{\perp}\left(\frac{dI_{dissociation}}{dR}\right)\Delta H_{a}^{ dissociation}(T_{a},P)\end{split}  \tag{9}\]  
The diffusivity across the interface from the liquid water side (\(R\) = 0 A) to the hydrate side of the interface (\(R\) = 12 A) is not known and is difficult to model using Molecular Dynamics simulations or other approaches. The profile in Figure 2 is, therefore, to be considered empirical, although it contains some interpretations of water movements during the dissociation of hydrates. It is considered sufficient for the purposes of this work. A mathematical fit to the profile in Figure 2 is  
\[-\log_{10}\left[\frac{D(R)_{j}}{D_{liquid,j}}\right]=\sum_{i=1}^{9}a_{i}\left( a\tan\left[\left(0.5R\frac{\pi}{2}\right)/12\right]\right)^{i-1} \tag{10}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Material** & **Thermal Conductivity (Experimental Values) W/(m\(\cdot\)K) 1** \\ \hline \multicolumn{2}{c}{Aquas Phase} \\ Liquid Water (273 K–283 K) & 0.56–0.58 \\ Sea Water & 0.56–0.57 [35] \\ CH\({}_{4}\) (liq) & [36] \\ CO\({}_{2}\) (liq) & [37] \\ \hline \multicolumn{2}{c}{Gas Phase} \\ CH\({}_{4}\) (gas) & 0.0297 (260 K, 1 MPa) 0.035(at 4.98 MPa, 277 K) \\ CO\({}_{2}\) (gas) & 0.093512 [37] \\ \hline \multicolumn{2}{c}{Minerals} \\ Quartz & 6.6–8.4 [38] \\ Calcite & 3–4.5 [38] \\ Kaolinite & 1.8–3 [38] \\ \hline \multicolumn{2}{c}{Hydrate Phase} \\ CH\({}_{4}\) Hydrate & 0.57 (263 K) [33] 0.62 [32] 0.68 (261.5–277.4) [33] \\ CO\({}_{2}\) Hydrate & 0.49 [34] \\ \hline \hline \end{tabular} 1  
\end{table} 
Table 2: Thermal conductivity of different hydrate components in various phases. Figure 2: Relative diffusivities of CO\({}_{2}\) across the interface from liquid water side to hydrate interface side. \(j=\text{CO}_{2}\), \(\text{CH}_{4}\), with parameters given in Table 3.  
However, in the context of this work, the most important aspect is to shed more light on the relative importance of various contributions to the implicitly coupled mass and heat transport dynamics related to hydrate phase transitions. [PERSON]'s law will give:  
\[J(R)=\left[-D_{\text{CO}_{2}}(R)\frac{\partial\mathcal{C}(R)}{\partial R}\right] \tag{11}\]  
while the derivative of mass flux across the interface can be written as  
\[\frac{dJ(R)}{dR}=\left[-\left(\frac{dD_{\text{CO}_{2}}(R)}{dR}\right)\left( \frac{\partial\mathcal{C}(R)}{\partial R}\right)-D_{\text{CO}_{2}}(R)\frac{ \partial\mathcal{C}^{2}(R)}{\partial R^{2}}\right] \tag{12}\]  
The appropriately derived concentration profile of \(\text{CO}_{2}\) across the water interface is illustrated in Figure 3 below: