## 4 Results and Discussion
As an illustration of the slow transport through hydrate films, we will utilize a simple version of CNT for spherical hydrate cores. CNT can be formulated as  
Figure 1.2  
where \(J_{0}\) is the mass transport flux supplying the hydrate growth. For the phase transition in Equation (1), it will be a supply of CO\({}_{2}\) across an interface of gradually more structured water towards the hydrate core, as discussed in [PERSON] et al. [29] and illustrated above (see Figures 2 and 3). The units of \(J_{0}\) will be mol/m\({}^{2}\)s for heterogeneous hydrate formation on the growing surface area of the hydrate crystal. \(\beta\) is the inverse of the gas constant times temperature, and \(\Delta G^{Total}\) is the molar free energy change of the phase transition. This molar free energy contains two contributions: the phase transition free energy as described by Equation (1), and the penalty work incurred by pushing aside the old phases. Since the molar densities of liquid water and hydrate are reasonably close, it would be a fair approximation to obtain it as a product of molar free energy of the phase transition times the molar density of the hydrate times the volume of the hydrate core. The push work penalty term is simply given by the interface free energy times the surface area of the hydrate crystal. Using the underscore symbol to indicate extensive properties (in Joule):  
Figure 1.3  
For the simplest possible geometry of a crystal, a sphere with radius \(R\), we will obtain  
Figure 1.4  
where \(\rho_{N}^{H}\) is the hydrate molar density, and \(\gamma\) is the interfacial free energy between hydrate and the surrounding phase. Even if the hydrate core is floating on the water surface, one would expect small crystals to be covered by water on the gas side as well due to capillary forces facilitating the transport of water molecules from the liquid water side. Differentiating Equation (15) with respect to R and solving for the maximum free energy radius (the critical core size) yields the usual result:  
Figure 1.5  
where superscript * denotes critical nucleus radius. The critical radius for two different temperatures is given in Figure 11 below. For the temperature below the transition to liquid CO2, the trend is very similar to that observed in Phase Field Theory (PFT) modeling (see [39; 40; 41; 42] and the references therein). The transition into the liquid phase will dramatically increase the critical radius, although it will still remain at the nanoscale, except for the lowest pressures. Figure 12 presents nucleation times corresponding to heterogeneous hydrate formation from CO2 and water phase at 280 K and 285 K, with the nucleation time above the carbon dioxide transition point much higher at lower pressures. As noted earlier, diffusion through the initial hydrate film will be very slow. Since most estimates available in the literature are based on molecular modeling simulations  
Figure 1.6 Calculated critical radius for heterogeneous nucleation of hydrate from CO2 and liquid water as a function of pressure for 280 K (solid) and 285 K (dashed). Figure 1.7 Calculated nucleation times for heterogeneous hydrate formation from CO2 and liquid water. Solid curve is for 280 K, and dashed curve is for 285 K.  
using specific approximations and assumptions, at least 12 different values can be found in journal papers; they range between 10 and 15 m\({}^{2}\)/s and 10-17 m\({}^{2}\)/s. As long a hydrate is formed, it will be the lowest free energy phase for water. The hydrate will therefore decide the minimum level of CO\({}_{2}\) in the contacting aqueous phase. This means that hydrates can grow from CO\({}_{2}\) dissolved in water from its solubility limit and down to the CO\({}_{2}\) hydrate stability limit (see [PERSON] et al. [29] for more details). Hydrate formation from the aqueous phase side will preferentially occur towards the already established hydrate film due to the presence of adsorbed and structured water in contact with the hydrate surface.