## 2 Characterisation of the Structural Evolution of Rocks Following Exposure to Carbon Dioxide
### Surface Area Determination  
As mentioned previously, the specific surface area of rocks is a key void space descriptor for estimating key parameters such as geochemical reaction rates, with errors in the input descriptors leading to great uncertainty in parameter predictions. In general, gas adsorption data, analysed using either the Langmuir or Brunauer-Emmett-Teller (BET) models, are used to obtain surface area [61]. However, both these models assume that the surface is homogeneous, such that the interaction with the adsorbate is the same everywhere, whereas rocks are complex mixtures of different minerals. The homotattic (meaning isoenergetic) patch model was introduced in order to account for the effect of the chemical heterogeneity of such surfaces on adsorption [62]. This model treats the surface of the adsorbent as a patchwork of zones with different types of adsorption site (like a checkerboard in the case of two different sites), where, perhaps in the case of rocks, these zones correspond to different mineral grains. The model assumes that each of the individual patches is large, so that edge effects, caused by interaction with other zones, are negligible. The observed adsorption thus consists of a composite of the behaviour of the set of patches, such that  
\[V=V_{m}(\theta_{1}I_{1}+\theta_{2}I_{2}+\ldots+\theta_{i}I_{i}+\ldots) \tag{7}\]  
where \(I_{i}\) is the isotherm equation describing adsorption on the \(i\)th patch, and \(\theta_{i}\) is the fraction of the surface occupied by patches of type \(I_{i}\), such that the various \(\theta_{i}\)-values obey  
\[\theta_{1}+\theta_{2}+\ldots+\theta_{i}+\ldots=1. \tag{8}\]  
Most past studies using the homotattic patch model have only distinguished each surface patch by means of a different adsorption energy, and have utilised the model simply as a way to curve-fit experimental isotherms _a posteriori_ without using the fitted parameters to make any predictions regarding the other physical properties of the system. For example, [PERSON] et al. [63] only fitted Langmuir-type isotherm models, which have only monolayer capacity and adsorption energy parameters, even into the multilayer and capillary condensation regions of the isotherms, with the latter being the region where models based only upon van-der-Waals interactions alone are not valid. However, adsorbate interaction strength is not the only property of the particular surfaces within different patches that can be probed by adsorption isotherms. Natural rock surfaces are also typically rough at the molecular scale. The impact of roughness on adsorption is neglected in the simplest models of gas sorption, such as the standard ISO BET equation [64; 65]. Irregularities in the surface, overall, cause the space for adsorbing the second and subsequent layers of an adsorption multi-layer to decrease with distance from the surface, such that the capacity of each successive layer declines. For surfaces that possess the special property of self-similarity over multiple length-scales, which are therefore fractals, the decline in the adsorbate capacity of each layer is given by the following equation [66]:  
\[\frac{A_{i}}{A_{1}}=i^{2-d} \tag{9}\]where \(A_{1}\) is the area in the first adsorbed layer, \(A_{i}\) is the area in the \(i\)th adsorbed layer, and \(d\) is the surface fractal dimension (\(2\leq d\leq 3\)). As the degree of surface roughness increases, the fractal dimension increases from 2 (for a flat surface) towards 3. This effect can be incorporated into the standard BET model resulting in a fractal version. This is given by [67]:  
\[log(v)=log(v_{m})+log\bigg{[}\frac{cx}{1-x+cx}\bigg{]}-(3-d)log(1-x), \tag{10}\]  
where \(v\) is the amount adsorbed, \(v_{m}\) is the monolayer capacity, \(c\) is the BET constant, and \(x\) is the relative pressure. The impact of the fractal roughness, as described by Equation (9), is to decrease the amount adsorbed for each successive layer of adsorbate.