# Numerical Phase-Field Model Validation for Dissolution of Minerals
## 3 Mathematical Methods  
### Analytical Solutions  
A following 1D planar analytical description of a diffusion-controlled dissolution for phase transformation is considered, where the solid is immersed in a (semi-)infinitive liquid solution. The diffusion equation is a parabolic partial differential equation, which is expressed as follows:  
\[\frac{\partial c(x,t)}{\partial t}=D_{\text{L}}\
abla^{2}c(x,t), \tag{3}\]  
where \(D_{\text{L}}\) is the diffusion coefficient, \(c(x,t)\) is the concentration at location \(x\) and time \(t\), subject to the conditions:  
\[c|_{x=R,\ t}=c_{\text{L}e}(0<t\leq\infty), \tag{4}\]  
\[c|_{x,\ t=0}=c_{\text{L}0}(x\geq R), \tag{5}\]  
where \(x=R\) is the position at the solid-liquid interface, \(c_{\text{L}0}\) and \(c_{\text{L}e}\) represents the initial concentration and the equilibrium concentration of one component in the liquid phase.  
At the solid-liquid interface, the following independent flux balance condition must be fulfilled:  
\[(c_{\text{S}}-c_{\text{L}e})\frac{dR}{dt}=D_{\text{L}}\frac{\partial c}{ \partial r}\Big{|}_{x=R}, \tag{6}\]  
where \(c_{\text{S}}\) is the concentration in solid phase which is taken as a constant.  
Figure 1: Schematic representation of solute concentration of soluble minerals in situ (i.e., green dotted line) and equilibrium states (i.e., red solid line), and the phase transformation within a diffuse and sharp interface, respectively. For the sharp interface, the evolution of the solute concentration is discontinuous at the interface. However, for the diffuse interface, the solute concentration evolves continuously between their equilibrium values at the mineral (\(c_{\text{Se}}\)) and solution boundary (\(c_{\text{L}e}\)).  
The exact analytical solution for the field is [6]:  
\[c(x,t)-c_{\rm M}=(c_{\rm Le}-c_{\rm L0})\frac{erfc\Big{[}(x-R_{0})/2\sqrt{Dt} \Big{]}}{erfc(-\lambda)}. \tag{7}\]  
The interface position at current time can be expressed as  
\[R=R_{0}-\lambda_{I}\sqrt{Dt}, \tag{8}\]  
where \(R_{0}\) denotes the value of \(R\) at the time \(t=0\), while  
\[\lambda_{I}=2\lambda, \tag{9}\]  
where \(\lambda\) is given by:  
\[\sqrt{\pi}\lambda\exp(\lambda^{2})erfc(-\lambda)=\beta/2, \tag{10}\]  
where \(\beta=2(c_{\rm Le}-c_{\rm L0})/(c_{\rm S}-c_{\rm Le})\).  
Different from the planar solid where exact solution is available, only approximately analytical solution model for the diffusion-controlled dissolution of the spherical solid has been found. The stationary-interface approximation is expressed as [6]:  
\[c(x,t)-c_{\rm M}=\frac{(c_{\rm Le}-c_{\rm L0})R}{x}erfc\left[\frac{x-R}{2(D_{ \rm L}t)^{1/2}}\right], \tag{11}\]  
where the current interface position is \(R^{2}=R_{0}^{2}-\beta D_{\rm L}t\); \(\beta\) as defined in Equation (10).  
The implicit expression for the particle radius ratio \(y\) (\(y=R/R_{0}\)) with respect to time is defined as follows:  
\[\ln[y+2p(\tau)^{1/2}y+\tau]+\frac{2p}{\left(1-p^{2}\right)^{1/2}}\mathrm{ arctan}\left(\frac{\left(1-p^{2}\right)^{1/2}}{\frac{y}{\tau^{1/2}}+p}\right)=0, \tag{12}\]  
where,  
\[\tau=\frac{\alpha^{2}t}{R_{0}^{2}}, \tag{13}\]  
\[\alpha^{2}=\beta D_{L}, \tag{14}\]  
\[p^{2}=\frac{\kappa}{4\pi}, \tag{15}\]