# Numerical Phase-Field Model Validation for Dissolution of Minerals
## 6 Results and Discussion  
### PF Validation against Analytical Solution for a Dissolution of Planar Mineral  
Under diffusion-controlled dissolution conditions, if the initial thickness of the diffuse interface (\(l_{0}\)) is known, the interface mobility \(L\) can be determined using Equation (A10). Due to the lack of relevant experimental data upon the values of \(l_{0}\), a parametric study of \(L\) was carried out. Under the assumption of a thin interface condition [48], the value of \(l_{0}\) should be taken much smaller than the minimum radial dimensions of the initial solid phase; however, from a computational point of view, it is expected that the thickness of the interface has to be as large as possible in order to keep the interface from being overly densely meshed, which increases the computational effort. Therefore, three cases of initial interface width, i.e., \(1\times 10^{-8}\) (PFM1); \(1\times 10^{-5}\) (PFM2) and \(1\times 10^{-4}\) (PFM3), were tested, corresponding to 0.0003%, 0.33% and 3.33% of the initial length of the solid phase. In addition, there must be at least 5 to 10 grid points in the interface area to ensure the stability of the numerical calculation and the reliability of the results [28]. Three cases of \(V_{0}\) (\(1\times 10^{-6}\) (PFM4); \(1\times 10^{-8}\) (PFM5) and \(1\times 10^{-10}\) (PFM6)) were tested. Figure 4 shows a comparison between the analytical (diffusion limited) model and six cases of PF models, where the reaction rates are slower than in case of diffusion control. As \(l_{0}\) decreases, \(L\) increases (as they are inversely related by Equation (A10)), causing a faster dissolution reaction, till reaching a limit defined by a diffusion control. At 8 h, PFM1 dissolves at a thickness 1.15 times greater than that of PFM3. The result of PFM2 is in good agreement with that of the analytical model. It can also be seen that the slope of the dissolution curve becomes progressively smaller with dissolution time due to the diffusion-controlled dissolution, i.e., the overall rate of dissolution slows down as it is governed by the diffusion flux and thus dependent on the concentration gradient that reduces with saturation of the solution. However, the (slow) reaction-controlled mechanism (see curves PFM4, PFM5 and PFM6) approaches linearity as the \(V_{0}\) decreases, and the lower the \(V_{0}\), the slower the dissolution speed. The above results show that the PF model developed is capable of describing both the diffusion-controlled and the reaction-controlled dissolution. Under the thin interface limit condition, the agreement between the analytical (diffusion-controlled) results and the PF model for the diffusion-controlled dissolution is in satisfactory agreement with the converged solution (PFM2). Figure 4: Comparison of analytical and phase-field (PF) model results of a dissolving planar mineral with variation of interface mobility \(L\), to shift from the fastest diffusion controlled mechanism towards the slower ones limited by reaction rates. A slight overestimation by the PFM1 model may be attributed to the used approximation (Equation (A10)) to numerically approach the diffusion-limited case. In this approximation, \(V_{0}\) is approximated as \(D_{\text{L}}/l_{0}\) (where \(D_{\text{L}}\) is the diffusion coefficient of the solute in the solution). The slight difference could also be the result from small incompatibility issues between the employed thermodynamic parameters, namely the used NaCl interfacial energy in PF model, and the NaCl solubility constant (\(p\)). Overall, we argue that the obtained agreement is overwhelming considering that no fitting calibration of the parameters has been performed. Figure 5 shows the spatial distribution of solute normalised concentration over time.