# Numerical Phase-Field Model Validation for Dissolution of Minerals
## 7 Conclusions  
Based on the results of this study, the following conclusions can be summarized:  
* by comparing with the results of the analytical method, it is verified that the PF model can accurately handle the dynamic evolution of the general diffusion-controlled phase transformation process;
* using NaCl as an example, the PF model can successfully simulate the mesoscopic evolution of inorganic non-metallic materials caused by diffusion-controlled dissolution. Using the derived interfacial mobility, the PF numerical simulation results show accurate and consistent agreement with the analytical method results, as well as with the experimental ones derived with video-microscopy images analyses. It is worth mentioning that all the input parameters of the PF model have real physical meaning and are based on the experiments data;
* an observed discrepancy was related to the dissolution mechanism, which was found to be initially limited by the reaction rate, being slower than the diffusion flux due to the rapid change of solute concentration. This change in dissolution mechanism was successfully captured by adjusting the PF interface mobility (\(L\)). * the dissolution characteristics of NaCl particles with different circularity factors were analysed by the 2D PF model. The simplification of spherical shape for NaCl particles was verified to hold. In future studies, the reaction control and diffusion control mechanisms will be combined with the second law of thermodynamics and non-equilibrium thermodynamics with respect to \(L\), so that \(L\) can be represented as the function of solute concentration or the Gibbs free energy of the reaction. In addition, numerical simulations need to be implemented at higher space dimensions to allow the introduction of complex microstructures in mineral  
Figure 13: Change in solute at the solution boundary with time. particles, such as pores, grain structure and surface roughness, so that their impact on dissolution kinetics can be assessed. The above results confirm that the dissolution kinetics of mineral particles can be successfully simulated using the employed PF model and the irregular morphological evolution can be effectively simulated in 2D. It is worth noting that the surface morphology of irregular particles has a strong influence on the dissolution kinetics. The complex evolution of particle morphology in physicochemical processes can be accurately evaluated only in a full 3D system. Furthermore, a dynamic (apparent) diffusion coefficient should be explicitly taken into account, e.g., as a function of concentration, which is of critical importance in analyzing the diffusion-controlled dissolution through porous materials involving additional chemical interactions. However, highly soluble (NaCl) crystals, as investigated in this work, can be considered as non-porous which leads to assume that their diffusion coefficient can be kept constant. It is worth mentioning that the proposed PF approach can be also extended for simulating the opposite processes of those presented in this paper, namely the precipitation of NaCl. This is part of forthcoming research and will be undertaken by enriching the current free energy density function of the liquid phase through adding an extra precipitation term, i.e., \(\Delta_{f}\). This should be considered for further development to develop a comprehensive mineral dissolution and precipitation modelling tool. Conceptualization, S.Y. and N.U. ; data curation, S.Y. and N.U. ; writing--original draft preparation, S.Y. ; writing--review and editing, everybody; supervision, N.U. and A.C.; project administration, N.U. and E.K. ; funding acquisition, N.U. and E.K. All authors have read and agreed to the published version of the manuscript. This research was funded by National German DFG organization under project number 426807554 titled \"Experimentally supported multi-scale Reactive Transport modeling of cementitious materials under Acid attack (ExpeRTa)\". No applicable.