# Numerical Phase-Field Model Validation for Dissolution of Minerals
## 6 Results and Discussion
### PF Validation against Experimental Results and Analytical Solution  
Figure 12 shows a comparison of the analytical solution, the PF model results and experimental results regarding the dissolution rate of individual NaCl crystals. The undissolved area is calculated from the residual solid phase length in the 1D simulation. The analytical solution is slightly below the lower boundary of the experimental values. In order to ensure that the dissolution reaction rate is completely controlled by the diffusion, the length (volume) of the exposure solution must be large enough so that semi-infinitive conditions are met, corresponding to the analytical solution. In that case no increase in concentration of solute should occur at the system (solution) boundary point. The length of solution 0.3 times (PFM_D1), 0.4 times (PFM_D2) and the original length (PFM_D3, 4.92 \(\times\) 10\({}^{-3}\) m) were tested using the PF method. From the results it can be seen that the dissolution rate slows down and the solute concentration increases significantly at the solution boundary point as the solution phase length becomes shorter (Figure 13). The change in solute at the boundary point, calculated using the solution lengths in the experiment (PFM_D3), is almost zero. It was thus verified that the diffusive dissolution of individual NaCl crystals can be simulated well using our implementation of the PF method.  
Figure 11: Phase profiles of single NaCl spherical particle along radial direction with time.  
Figure 12: Comparison of numerical and experimental results of NaCl single particle dissolution.  
Figure 10: Concentration profiles of single NaCl spherical particle along radial direction with time.  
The experimental result in Figure 12 shows lower initial rates, due to the reaction controlled mechanism. Initially, the diffusion flux is very high, as the solute concentration of the initial exposure solution is zero (initial condition). Therefore the overall reaction rate is limited by the reaction rate which is lower than the initial high diffusion flux. Such reaction rates in the PF model can be considered in the interface mobility (\(L\)) that should vary with time and is fundamentally a function of the Gibbs free energy of the chemical reaction or the solute concentration. In future, it should be attempted to physically represent the interface mobility kinetics (\(L\)) as a function of the solute-under saturation, as commonly used in reaction rates expressions. Such a more fundamental approach is still missing in PF, due to the complexity in its mathematical derivation. Here, \(L\) is adjusted in a simplified way as a smaller value (4.11 \(\times\) 10\({}^{-6}\)) within 100 s and a larger value (4.11 \(\times\) 10\({}^{-3}\)) after 100 s whose result is represented by PFM_R1. As can be seen from the comparison, the dissolution rate is relatively flat at the early stage of dissolution when process controlled by the reaction. The curve of PFM_R1 is higher than that of PFM_D1, D2 and D3. However, after 100 s, PFM_R1 almost overlap with the other three due to the conversion of the dissolution rate control into the diffusion control mechanism.