# Numerical Phase-Field Model Validation for Dissolution of Minerals
## 1 Introduction  
Mathematical modelling of the moving-boundary dissolution fronts of minerals is important in a wide range of engineering technologies. For example, it is of great importance in fields of geochemistry, materials science, hydrometallurgy, etc. Predictions of the moving boundary dissolution phenomena can support in the design of engineering processes where dissolution is desired: e.g., in extraction of elements or reactivity of cementitious minerals, but also when not desired, e.g., in durability (corrosion) issues of building materials (e.g., steel-reinforced concrete frames). In general, minerals dissolve when exposed to aggressive solution environments and form leached layers of varying density and strength [1]. This in turn affects the mechanical and transport properties of the microstructure which further may be relevant at higher scales, for example when the material (rock, concrete or mortars) has structural applications. In addition, the dissolution mechanisms of some special minerals can be of great industrial and environmental interest. For example, the dissolution of scorodite is considered a potentially good carrier for arsenic fixation [2]. Moreover, the application of innovative self-healing concrete in civil engineering has been extensively and intensively studied in recent years. The dissolution of Ca(OH)\({}_{2}\) from the concrete matrix is one of the key processes of the durability and self-healing mechanisms [3, 4, 5]. The dissolution of minerals often involves complex physico-chemical processes at the solid-liquid interface. However, this can be simplified at the mesoscale to the problem of a continuously moving boundaries. Traditional sharp interface models are thus requiredto trace these moving fronts [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. However, this becomes extremely difficult for high-dimensional problems, with complex dynamic geometries, especially those whose interface evolution is accompanied by energy changes. In this sense, phase-field (PF) methods provide a powerful way to track such interfaces. The PF method has been applied to various phenomena in materials science area, such as, solidification, solid-state phase transformation, recrystallization, grain growth, fracture, and electromigration [17; 18; 19; 20; 21; 22]. Generally speaking, the PF model can be regarded as a kind of diffusion interface model [23; 24; 25], which assumes that the thickness of the interface is limited, while the physical properties on the interface are continuous and smooth [26; 27]. The moving solid-liquid interface can thus be accurately tracked [28]. Compared with sharp interface models, the PF diffusion interface model has the important advantage that no boundary conditions are specified on the interface between different domains [29; 30]. This allows us to study the evolution of arbitrarily complex morphology without tracking the microscopic shape of the grain [31; 32]. A new variable, namely the order parameter, is required to represent the ordered numbers of materials in terms of time and position [33; 34]. In these works, the complex interfacial spatial-temporal evolution has been investigated through the aforementioned order parameter in an implicit way. An additional feature of the PF method is that there is a functional total free energy that can characterize the nature of the phase transition. It includes the various energy contributions of the system at the non-equilibrium state: i.e., chemical energy [35; 36; 37; 38], electric potential energy [39; 40; 41], stress energy [42; 43; 44], etc. It is the competition between these different energies that leads to the generation of changing microstructural topography during the phase transition. In solving the PF model, conservative fields such as concentration fields can be described e.g., through the Cahn-Hilliard equation [45], while non-conservative fields, as order parameters, can be described via the Allen-Cahn one [46].