# Numerical Phase-Field Model Validation for Dissolution of Minerals
## 6 Results and Discussion
### The Effect of Mineral Shape: Dissolution Simulation by Two-Dimensional (2D) PF Model  
Irregular particles can be simplified by spherical shapes provided certain conditions are met. Numerical simulations are then performed using the spherical symmetry and  
Figure 5: One-dimensional distribution of solute concentration by PF model (\(L=4.02\times 10^{-1}\)). Figure 6: One-dimensional distribution of interface by PF model (\(L=4.02\times 10^{-1}\)). coordinate system, reducing the model to only one space dimension (1D model). However, some studies have shown, both theoretically and experimentally, that the particle shape and surface roughness may affected the dissolution rates [98; 99; 100; 101; 102]. Therefore, before adopting the spherical simplified 1D model for NaCl crystals in this study, the effect of particle shape on the dissolution rate was analysed. In this way one can determine whether the simplification of the spherical shape for NaCl particle having some circularity factors (0.71 \(\pm\) 0.06) as in used literature data [95] is reasonable. Therefore, Figure 7 shows the 2D dissolution simulation results over time for spherical, elliptical and irregular shapes but with the same area. Qualitatively, the sharp edges of irregular shapes gradually disappear during the initial stages of dissolution, and the curvature decreases until they are completely rounded. The ratio of the major axis to the minor axis of the ellipse gradually decreases and develops towards the circular direction. The radius of the circle decreases gradually and the curvature remains constant. It can be seen that the dissolution under all shapes follows the process of spheroidization. The edges of the particle become smoother during dissolution. As the dissolution reaction proceeds, the morphological differences between the particles with different shapes become smaller. The reason for this trend is that among closed geometries of equal area, circles have the smallest circumference. This means that the total interfacial free energy of the solid-liquid is minimal. Irregularly shaped solids have a high solid-liquid total interface free energy due to their uneven boundaries. A high interfacial energy means a high total free energy of the system. The system always tends to decrease the total free energy, this being an important factor in determining the mineral shape during dissolution. The area of the solid-liquid interface tends to decrease, which causes the flange at the solid-liquid interface to disappear and eventually to become round. Since the solid-liquid interface is described in the PF model as a diffuse interface, with a certain width, it is difficult to describe the interface length in terms of the absolute perimeter of the solid phase. However, in order to characterize the change in the surface morphologies of crystals for different shapes with time, the contour length of \(\phi\) = 0.5 is taken to approximate the solid-phase perimeter. Figure 8 shows that at the initial condition, the irregular mineral grain has the maximum interfacial length. As the dissolution reaction proceeds, the interfacial length gradually decreases and the curves of the ellipse and circle almost coincide. Figure 7: Snapshots corresponding to different time points in the dissolution profile of the NaCl crystal in 2D simulation. The normalized phase ratio \(\phi^{*}\) is defined as \(\phi^{*}=(\phi_{t}-\phi_{min})/(\phi_{max}-\phi_{min})\), being \(\phi_{t}\) the integration of the phase at the time t, over the domain, while \(\phi_{min}\) and \(\phi_{max}\) are representing the min and max integration of the phase, respectively (for \(c\) as well). The profile of the normalized phase ratio physically represents the projected area of particles. It can be seen from Figure 9 that the projected area of the three shapes of particles decreases with time, while the normalized concentration ratio (\(c^{*}\)) keeps constant, which proves the conservation of mass for solute transport. The slope of \(\phi^{*}\) becomes progressively smaller.