## 5 Mathematical Models  
CS may be used to describe the dynamic behavior of physicochemical systems using simulations and models. Mathematical models can be created at the three conventional scales: microscopic, macroscopic, and kinetic [91].  
On the microscopic scale, the state of the system is made up of all of its component states and is solely a function of time. The dynamics are modeled using a large system of ordinary differential equations, each representing a variable at the microscopic level. At the macroscopic level, continuum models for systems whose state variable can be a vector, are created. This kind of variable, which depends on both space and time, is provided by locally averaged data. Partial differential equations derived from equilibrium or conservation rules characterize the dynamics within the elementary volume of the space of microscopic states. At the mesoscopic scale, kinetic theory models can be also defined for systems composed of numerous interacting units, each of whose unique states is referred to as a microscopic state. The dynamics are described by a differential system created by conservation relations within the elementary volume of the space of microscopic states [91]. Furthermore, mathematical models at different scales can also be combined to form a hybrid technique called a multiscale approach [92].  
Although the idea of multiscale modeling impacts many domains, material modeling has only benefited greatly from it in recent decades. Since the physics at a finer scale is better understood than at a coarser size, multiscale modeling refers to formulating equations, parameters, or simulation techniques that characterize behavior at a particular length scale starting from information obtained at the finer scale [92]. Many of these techniques have been created, employing different strategies to bridge various lengths and time spans. For a summary of multiscale computational methods and representative examples, the reader may refer to Table 1 of the work by Fish et al. [92]. As examples, we provide below two indicative samples taken from that table:  
_Upscaling methods based on the same mathematical models:_ On different scales, the same (or similar) physics models are applied. The interscale coupling could be based on different techniques such as asymptotic theory, wavelets, or macrohomogeneity conditions [92]. _Resolved-scale methods based on a Multigrid approach:_ Separate physics models and discretization schemes are used in different spatial domains that can be simulated concurrently. Coupling strategies are based on interscale operators (restriction and prolongation). The solution algorithms are similar to iterative Multigrid matrix solvers [92].  
The extensive treatment of the \"multiscale approaches\" goes beyond the scope of this work but could be the objective of future developments. In any case, for those wishing to delve deeper into the topic, they can refer to the interesting review by Fish et al. [92].  
According to what has already been tested and described in the literature cited in Section 3, we will only focus on models used at both macro and mesa scales, neglecting models that work at micro-scales.