# Numerical Phase-Field Model Validation for Dissolution of Minerals
## 1 Introduction
There are four models in the literature that are most commonly used to address two- or multi-phase transition processes. They provide important insights into solving of the mineral dissolution problems: the Wheeler-Boettinger-McFadden (WBM) model [30; 47], the Kim-Kim-Suzuki (KKS) model [48], Steinbach model [49] and [PERSON] model [50]. The WBM model is derived in a thermodynamically consistent way, which is based on an assumption that each point of the interface is a mixture of coexisting phases with the same composition but different volume fractions [27]. This model works under both sharp-interface condition and finite-interface thicknesses [51; 52]. However, the larger interface thickness will lead to unreliable calculation results [53; 54; 55]. KKS model shows a different definition of the free energy density, which defines the interface as a mixture of liquid and solid phases of different compositions, but with the same chemical potential [56]. For the KKS model, the relationship between model parameters and material properties can be established through the equilibrium and thin interface limit analysis [48]. The Steinbach model is not based on the thermodynamic treatment, but based on the geometric description of interface through the interpolating function of interface curvature. This model is more suitable for a dilute alloy [57]. [PERSON] et al. [50] finally used the similarity of alloys and pure materials to expand the thin interface model by matching variables in pure materials. However, there are two strict assumptions in the model that limit its application (1) the liquidus and solidus lines need to be parallel, and (2) the diffusivity of the solute is constant in the entire region [58]. In applying the PF model, described above, to the moving boundary problem of minerals, it is necessary to understand how to select the interface mobility so that the model can effectively describe the dissolution process. In the literature, there are only a few studies that address this issue. [PERSON] and [PERSON] [59] proposed that, in a single-component system, the interface mobility is related to the interface velocity and the driving force according to the chemical rate theory. In the case of spinodal decomposition, the interface mobility can be obtained from the diffusion coefficient and thermodynamics. When the model is used to simulate the complex meso-morphological evolution, the interface mobility needs to be determined based on experiments, as also demonstrated in this paper (Section 6.3). [PERSON] and [PERSON] [25] made a linear approximation of the temperature gradient at the interface and proved that in dealing with the solidification problem of pure melting, the PF parameters can be accurately determined under the thin interface limit. Based on this model, [PERSON] and [PERSON] [60] developed a phase-field approach for aqueous dissolution/precipitation reactions assuming first order reaction kinetics. The model was validated by a one-dimensional analytical solution of interface motion due to solute precipitation. Two additional terms were added to the diffusion equation, one corresponding to the discontinuity of the solute concentration gradient, at the interface, while the second one represents the net source (or sink) of the solute, coming from the discontinuity in the solute concentration across the interface. In most of the studies, the values of interface mobility are used as empirical or hypothetical ones [61; 62; 63; 64; 65; 66; 67]. Furthermore, few attempts have been made to explain in detail the calculation of the interface mobility and its relation to other physical parameters [51; 68; 69; 70; 71]. Therefore, tackling of this difficulty will be one of the innovations of this paper. In contrast to the reaction kinetics controlled case, here we further validate the PF approach on the experimental results (of NaCl dissolutions) and are focusing mainly on the diffusion limited mechanisms.