# A Physical Approach to Simulate the Corrosion of Ceramic-Coated Magnesium Implants
## 1 Introduction  
Bone fractures have a high incidence and are treated by osteosynthesis, which includes implanting fixation devices to hold the adjacent bone fragments still until the completion of the healing process [1]. The market size for these devices in the United States reached $6.9 billion in 2017 [2]. Bio-inert metals with high corrosion resistance such as stainless steel and titanium alloys have been the conventional materials used to make these fixation devices due to their strength and biocompatibility. The downfall is that another surgery is usually needed to remove the implant after healing as they become redundant and may cause severe complications. In other cases, where bone defects are difficult to treat by osteosynthesis alone, bone grafting techniques are used [3]. In auto-grafting for instance, bone tissue is extracted from a different healthy bone to fill the defected site and provide support to the healing bone tissue [3]. However, it becomes challenging as the defect size increases. Therefore, bioabsorbable alternatives grabbed the attention in the past two decades as base materials for osteosynthesis devices as well as bone scaffolds for bone grafting. Magnesium alloys are given more research interest due to their superior biocompatibility and closest range of mechanical properties compared to natural bone tissue and other used metals for orthopedic implants (see Figure 1) [4]. However, magnesium is the least noble metal in the galvanic series with an electrode voltage of \(-2.372\) V against standard hydrogen electrode [5]. Hence, alloying and surface treatments were investigated in the literature to improve the corrosion resistance and strengths of pure magnesium [6]. Moreover, ceramic coatings made by the promising Micro Arc oxidation coating method have been shown to improve the corrosion resistance significantly for magnesium and its alloys [7; 8; 9]. As a result, the corrosion behavior of magnesium-based implants is challenging to predict in an aqueous environment, and it requires extensive in vitro and in vivo corrosion experiments to best predict this behavior. In order to provide an inexpensive method to predict this corrosion behavior, numerical simulations have been introduced over the past decade to describe the different corrosion mechanisms found in magnesium alloys. For instance, a phenomenological approach based on the continuum damage theory was used to describe the uniform corrosion as well as the heterogeneous pitting corrosion combined with the stress corrosion cracking [10; 11; 12; 13]. Physical modeling of the corrosion was also used to model a reaction-controlled corrosion behavior as well as transport-controlled corrosion using diffusion law [14; 15; 16; 17; 18]. In the model by [PERSON] et al. [18], a transport-controlled corrosion was assumed to carry out the model for a uniform corrosion behavior in pure magnesium. The transport was assumed mainly diffusive and follows [PERSON]'s law of diffusion in Equation (1). \[\frac{\partial c_{Mg}}{\partial t}=\
abla\big{(}D_{Mg}\;\
abla c_{Mg}\big{)} \tag{1}\]  
where \(c_{Mg}\) is the magnesium ions concentration, and \(D_{Mg}\) is the diffusion coefficient of the magnesium ions in the environment, which was assumed isotropic. In this model, diffusion is assumed following the saturation of the ions near the corrosion surface. Subsequently, the corrosion surface is moved inward to model the loss in the volume due to corrosion. The surface movement was modeled in an arbitrary Lagrangian-Eulerian adaptive mesh and finite element method. [PERSON] et al. extended the diffusion-controlled model by introducing a coupled system of partial differential equations to capture the interaction between the corrosion products and chloride species with magnesium ions [14]. The concept of the effective diffusivity of the magnesium ions through the corrosion products was also introduced. The level set method was used to model the corrosion surface movement in a 2D finite element model.