# Numerical Modelling of Effects of Biphasic Layers of Corrosion Products to the Degradation of Magnesium Metal In Vitro

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}and MgCO

_{3}; this is the first study to consider the latter product in a magnesium corrosion model. A further novelty is to consider these layers as porous media, whereby there is fluid phase flow within the pores of the developing crystal structures, so that the reactants H

_{2}O and CO

_{2}can advect, as well as diffuse, through them. Whether or not this porous media assumption leads to substantially different results will be one of the aspects explored in this paper. An aim is to guide relatively simple in vitro experimentation that can inform the model parameters, which can then be used in the modelling of magnesium in more clinically relevant environments, with more appropriate geometries and dimensions, to predict corrosion in vivo.

## 2. Mathematical Model

_{2}reacts with Mg(OH)

_{2}to ultimately form magnesium carbonate in a reaction summarised by [25,26],

_{2}are hydrogen carbonates ions, HCO

_{3}

^{−}, formed from the reaction of dissolved CO

_{2}with water. The reaction of these ions with Mg(OH)

_{2}leads to the formation of magnesium hydrogen carbonate, Mg(HCO

_{3})

_{2}, which then decomposes to form magnesium carbonate, MgCO

_{3}. The intermediate magnesium hydrogen carbonate is thermodynamically unstable at atmospheric levels of CO

_{2}[26] and demonstrated experimentally [7,27]; we thus assume the intermediate hydrogen carbonate form is short-lived and will therefore be neglected in the modelling. A more realistic representation of the overall reaction is [7],

_{3}

^{−}will lead to the corrosion of Mg(OH)

_{2}via the reaction in Equation (4). The formulation of the model with regards to conversion of the hydroxide to carbonate forms means that both reactions Equations (2) and (4) are described, and the variable ${C}_{2}$ in this model can be viewed either as the concentration of CO

_{2}or HCO

_{3}− or both as the stoichiometry for water is the same; for simplicity, the discussion on Mg(OH)

_{2}corrosion in the remainder of this paper will refer to CO

_{2}and the reaction in Equation (2). We note that the resulting layer of magnesium carbonate is more stable and has been proposed as a layer to delay the corrosion process [25]. We further assume in the model that, throughout the corrosion process of Mg, the environment is stable (e.g., pH is unchanged, as would be expected in a buffered medium in vitro) and that supplies of water and CO

_{2}are inexhaustible.

_{2}, which in turn is surrounded by a layer of MgCO

_{3}(see Figure 1). In order for the magnesium pellet to corrode further, water must be able to diffuse though the carbonate and hydroxide layers to react at the pellet’s surface and CO

_{2}must be able to diffuse through the carbonate layer to reach the hydroxide compound interface. These assumptions lead to a model that describes both the transport and reaction processes of water and carbon dioxide as well as the location of the interfaces between magnesium and its compounds, which are deposited on the surface of the magnesium metal as corrosion products. The modelling will be formulated for a general 1D geometry, namely Cartesian (describing a magnesium slab), cylindrical (a magnesium rod) and spherical geometry (a magnesium ball). The hydroxide and carbonate layers are treated as porous media, thereby the movement speed, ${\mathit{v}}_{{\mathit{s}}_{\mathit{i}}}$, of the “solid” components, i.e., the Mg(OH)

_{2}and MgCO

_{3}, is distinct to that of the fluid and dissolved gas components, i.e., H

_{2}O and CO

_{2}, namely ${\mathit{v}}_{\mathit{f}}$; this is a novel feature in metal corrosion models. Fortunately, by assuming ideal geometries, a closed system of equations can be derived based on mass conservation alone.

#### 2.1. Mathematical Modelling

- $r=\alpha \left(t\right)$ is the location of the magnesium to magnesium hydroxide interface.
- $r=\beta \left(t\right)$ is the location of the magnesium hydroxide to magnesium carbonate interface.
- $r=S\left(t\right)$ is the location of outer edge, exposed to concentrations of water and carbon dioxide, representative of the in vitro environment containing HCO
_{3}^{−}/CO_{2}buffering system.

- Zone 0: the Mg layer $r<\alpha \left(t\right)$,
- Zone 1: the Mg(OH)
_{2}layer $\alpha \left(t\right)<r<\beta \left(t\right)$, - Zone 2: the MgCO
_{3}layer $\beta \left(t\right)<r<S\left(t\right)$.

_{2}in vitro, we note that in the physiochemical representative corrosion system, the magnesium and magnesium hydroxide regions will eventually vanish, i.e., $\alpha =0$ and $\beta =0,$ respectively; consequently, there are distinct time phases in the corrosion process that need to be separately handled by the model. We define $t={T}_{\alpha}$ as the point in time when $\alpha \left(t\right)=0$ (i.e., $\alpha \left(t\right)>0$ for $t<{T}_{\alpha}$), and likewise $t={T}_{\beta}$ for when $\beta \left(t\right)=0$. Once $\beta =0$, i.e., for $t>{T}_{\beta}$, there are no further developments in the system and all that remains is a block of magnesium carbonate.

_{2}on the $r=\beta $ interface (see Section 2.1.1). Let ${W}_{1}$ be the mass concentration of water in the pores of Mg(OH)

_{2}structure and ${W}_{2}$ and ${C}_{2}$ be that of water and carbon dioxide, respectively, in the MgCO

_{3}. The transport equations for the water and carbon dioxide are

#### 2.1.1. Boundary and Interface Conditions

- Case 1 considers the limit $k\to \infty $, where the reaction is so rapid that water is immediately exhausted on $r=\alpha \left(t\right)$ interface, hence ${W}_{1}=0$ here. This assumption is most consistent with that used for carbon dioxide on $r=\beta \left(t\right)$. The boundary conditions Equations (13) and (15) are relevant.
- Case 2 considers $k<\infty $, whereby ${W}_{1}>0$ on $r=\alpha \left(t\right)$. In a short time, whereby $1-\beta \left(t\right)/S\left(t\right)\ll 1$, the small distance for the carbon dioxide to diffuse means that Mg(OH)
_{2}immediately becomes exhausted on production and Mg converts to MgCO_{3}, in effect, immediately; consequently, $\alpha \left(t\right)=\beta \left(t\right)$ during this transient. In time, the thickness of Zone 2, $S-\beta $, becomes sufficiently large for the reaction to exhaust the carbon dioxide on $r=\beta $, allowing the Mg(OH)_{2}layer to grow. Let $t={T}_{\alpha =\beta}$ be the smallest time at which ${C}_{2}(\beta \left(t\right),t)=0$; then, for $t<{T}_{\alpha =\beta},$ the conditions in Equation (16) hold, whilst, for $t>{T}_{\alpha =\beta}$, Equations (13) and (15) are then imposed.

_{2}is ${\omega}_{\alpha}-1$, where ${\omega}_{\alpha}={\mu}_{0}/{\mu}_{1}$; consequently, volume gain rate from the reaction yields ${v}_{{s}_{1}}A=-({\omega}_{\alpha}-1)\dot{\alpha}A$, noting that ${\omega}_{\alpha}>1$ implies a gain in volume so that ${v}_{{s}_{1}}$ must have an opposite sign to $\dot{\alpha}$. The final condition results from a no slip condition to the fluid phase on $\alpha \left(t\right)$, i.e., ${v}_{{f}_{i}}=\dot{\alpha}$. In summary, the conditions are on

_{2}, whilst a water molecule is produced; for the latter, we assume the concentration is continuous across the interface, i.e., $\left[{W}_{i}\right]=0$, where the shorthand $\left[{W}_{i}\right]={W}_{2}-{W}_{1}$ is used below. Letting A be again the area of a surface element on $r=\beta $, then the rate of volume loss of Mg(OH)

_{2}is $({v}_{{s}_{1}}-\dot{\beta})A$ and the molar loss rate is therefore ${R}_{\beta}={\mu}_{1}({v}_{{s}_{1}}-\dot{\beta})A$. Consequently, the difference in molar flux of the water is $\left[A(\dot{\beta}{W}_{i}-{J}_{{W}_{i}})\right]/{M}_{W}={R}_{\beta}$, and carbon dioxide is $A(\dot{\beta}{W}_{i}-{J}_{{W}_{i}})/{M}_{C}=-{R}_{\beta}$. The volume fraction difference from the reaction is ${\omega}_{\beta}-1$, by definition, and thus the volume gain rate $A({v}_{{s}_{2}}-{v}_{{s}_{1}})$ is equal to $({\omega}_{\beta}-1)({v}_{{s}_{1}}-\dot{\beta})A$. Conservation of fluid flux across the interface leads to $\left[(1-{\epsilon}_{i})(\dot{\beta}-{v}_{{f}_{i}})\right]=0$. The conditions are on

_{3}generates a volume fraction difference of ${\omega}_{\beta}{\omega}_{\alpha}-1$ and the stated condition on ${v}_{{s}_{1}}$ is equivalent to that in Equation (13) and likewise for ${v}_{{f}_{1}}$. The flux condition on water results from the net loss of one molecule from the overall reaction and likewise for CO

_{2}.

_{3}will have no exit from the system, so the final state corresponds to when $\beta =0$, whereby all of the Mg and Mg(OH)

_{2}have been exhausted.

#### 2.2. Exact Solutions

_{3}being ${({S}_{\infty}/{S}_{0})}^{1+d}={\omega}_{\alpha}{\omega}_{\beta}$.

#### 2.3. Non-Dimensionalisation

## 3. Numerical Method and Results

_{2}layer. We investigate in Section 3.3 the effect of the initial size of the block, which is, of course, an experimentally controllable parameter. In Section 3.6, the significance of the porous media assumption in the current model is examined. We note that all of the results shown are the dimensionless form of the variables, whereby one space unit represents 1 cm and one time unit represents about 14.5 h.

#### 3.1. Magnesium Degradation

_{2}takes $t=O\left({10}^{3}\right)$. This is largely due to a relatively low concentration of CO

_{2}compared to H

_{2}O in the fluid phase.

_{2}concentration distribution at the start of Phases 2.1, end of Phase 2.1 ($t={T}_{\alpha =\beta}$), Phase 2.2 at the point the full system is solved numerically (see Appendix B.3), the start of Phase 2.3 ($t={T}_{\alpha}$) and the end of Phase 2.3 ($t={T}_{\beta}$); the times t are detailed in the caption. In a short time, there is only a very narrow MgCO

_{3}layer present and, as expected, the water and CO

_{2}are very nearly uniform $r\in (\beta ,S)$. Furthermore, CO

_{2}is not initially exhausted by the conversion reaction of Mg(OH)

_{2}to MgCO

_{3}at $r=\beta $, but, in time, it descends, reaching zero as Phase 2.2 begins. As time advances, clear gradients in concentrations emerge and, whilst $\beta -\alpha $ and $S-\beta $ remain small, the concentrations of water and CO

_{2}appear linear. The upward kink in the water distribution is due to production at $r=\beta $ in the conversion reaction Equation (2), even exceeding the exterior concentration as, locally, water replaces CO

_{2}molecules. During Phase 2.3, when there is no more Mg remaining, the water distribution ${W}_{1}$ tends to a uniform distribution via diffusion and the zero flux condition at $r=0$. By the end of Phase 2.3, the profiles of ${W}_{2}$ and ${C}_{2}$ are no longer linear and CO

_{2}concentration forms a boundary layer in the vicinity of $r=\beta $.

#### 3.2. Effects of Geometry

_{2}and ultimately degrade into MgCO

_{3}, as can be seen from the final sizes, where ${S}_{\infty}^{\left[d\right]}$, ($d=0,1,2$ for the Cartesian, cylindrical and spherical geometry, respectively), ${S}_{\infty}^{\left[2\right]}<{S}_{\infty}^{\left[1\right]}<{S}_{\infty}^{\left[0\right]}$. Comparing the cylindrical case with that of Figure 2, we observe that, as expected, the magnesium layer disappears much faster in $\kappa \to \infty $ case (${T}_{\alpha}\approx 1$) than for $\kappa \approx 0.04$ (${T}_{\alpha}\approx 33$); but we note that it does not significantly affect the overall degradation time of Mg(OH)

_{2}. In reality, the limit $\kappa \to \infty $ is unlikely to be realistic for pure or mostly pure magnesium, and represents a metal of low purity. However, as degradation of Mg(OH)

_{2}is independent of $\kappa $, there is very little difference between the $t={T}_{\beta}$ values.

#### 3.3. Effect of Magnesium Block Size

_{2}being relatively small, so that $1\ll {\gamma}_{2}$, and hence the decay of the Mg(OH)

_{2}is slow. Here, $\dot{\beta}\ll 1$ and hence ${v}_{{f}_{2}}\ll 1$, and then Equation (25) would be expected to tend to the quasi-steady profile, with ${\partial}_{r}{r}^{d}{\partial}_{r}{C}_{2}\sim 0$, hence ${\partial}_{r}{C}_{2}(\beta ,t)\sim A/{\beta}^{d}$ for some constant $A>0$. Writing $r={S}_{0}\overline{r}$ and $\beta ={S}_{0}\overline{\beta}$, then, from Equation (30), we obtain

#### 3.4. Effect of Porosity of the Mg(OH)_{2} and MgCO_{3} Layers

_{2}layer, hence decreasing the rate at which water reaches the Mg interface. However, the solid fraction does not have an impact on the degradation time for Mg(OH)

_{2}as the transport of CO

_{2}in the MgCO

_{3}layer governs this process. The figure emphasises the importance of the hydroxide layer at slowing the degradation of the metal core by impeding the passage of water. The nonlinear relationship as predicted by the current model is in accordance with experimental data [34], whilst decreasing in porosity having the effect of enhancing longevity is consistent with Sun et al. [35].

_{3}does not appear to have an effect on ${T}_{\alpha}$, but does affect ${T}_{\beta}$. Here, for $t<{T}_{\alpha}$, the thickness of the MgCO

_{3}layer, $S-\beta $, is fairly small and appears not to be sufficient to impede significantly the passage of water across it, thus ${T}_{\alpha}$ remains approximately constant. Although, ${T}_{\alpha}$ varies between the $\kappa $ values, it is small compared to ${T}_{\beta}$, and the conversion of Mg(OH)

_{2}to MgCO

_{3}being independent of $\kappa $ means the plots are superimposed. As ${\epsilon}_{2}$ increases, it is CO

_{2}that is impeded by the smaller void fraction, leading to the sharp rise in time ${T}_{\beta}$ as ${\epsilon}_{2}\to 1$. Using the argument in Section 3.3 in formulating Equation (36) for large ${\gamma}_{2}$, we then expect ${T}_{\beta}\propto 1/(1-{\epsilon}_{2})$; this relationship matches the numerics very well, suggesting that ${T}_{\beta}\to \infty $ as ${\epsilon}_{2}\to {1}^{-}$.

#### 3.5. Effect of Rate of Reaction at Magnesium Interface

#### 3.6. Role of Advection

_{2}and MgCO

_{3}crystal structures allows transport of water and carbon dioxide through its pores. The separate treatment of the resulting fluid and solid phase velocities is in contrast to [16], in which they assumed that the transport of the diffusive species is supplemented by that of the solid phase motion; this presumably reflects these molecules being somehow connected to and dragged along by the crystal structure. Figure 9 compares the evolution of $\alpha ,\beta $ and S from three choices of advective flux velocities, ${V}_{i}$, of water and CO

_{2}, namely

- Case (i).
- The current model based on porous media assumption (${V}_{i}={v}_{{f}_{i}}$, solid lines).
- Case (ii).
- Zero advective transport (${V}_{i}=0$, dotted lines), i.e., ${v}_{{f}_{i}}$ set to zero in Equations (23)–(25) and in the boundary conditions.
- Case (iii).
- Advective transport equal to the solid phase velocity, as in [16] (${V}_{i}={v}_{{s}_{i}}$, dashed lines), i.e., ${v}_{{f}_{i}}$ swapped with ${v}_{{s}_{i}}$ in Equations (23)–(25) and in the boundary conditions.

## 4. Discussion

_{3}. The model was analysed numerically, but small time asymptotic solutions were needed to deal with singularities at initial and certain time points. In principle, the modelling approach is generic and can be used or adapted to model the corrosive process of any metals or alloys.

_{2}and MgCO

_{3}layers to close the system [36].

_{2}layer, $t={T}_{\beta}$. However, there is scope for these parameters to be estimated based on appropriate in vitro data. For example, data from time-course measurements of the proportion of constituents of small, spherical magnesium or magnesium alloy beads, immersed in appropriate media. The use of small beads, say around 0.1–1 mm radius (see Section 3.3), should ensure the experiment to be completed in a practicable and cheap time frame, whilst them being spherical enables direct application of the model to calibrate k, ${\epsilon}_{1}$ and ${\epsilon}_{2}$ with the data. The interpretation of k can be extended to the corrosion rate of different quality of magnesium metal and its alloy. In particular, the presence of impurity and the grain size of micro-structures, depending on the preparation methods used, can have dramatic effects on the corrosion rate together with environmental factors [37,38,39,40]. A further experiment involves the use of computed tomography (CT) images, which enables spatial details of the macroscale crystal structure that can be used to obtain direct measurement of ${\epsilon}_{1}$ and ${\epsilon}_{2}$. The model, with these tuned or determined parameters, provides a starting point to predict the corrosion properties of much larger magnesium pellets and in any 2D or 3D extension outlined above.

_{2}and MgCO

_{3}, so that the pure Mg is more exposed to the environment and accelerating its corrosion [11,41]. Furthermore, the tougher outer layer of MgCO

_{3}will itself be corroded and the resulting magnesium ions will eventually disperse and be excreted by the host; the modelling of this corrosion process leads to a modified boundary condition on $r=S$. Chloride ions in plasma will also react with Mg(OH)

_{2}to form MgCl

_{2}[9]; here, the model can be extended to consider two reactive species for Mg(OH)

_{2}, and assume, for simplicity, that the outer layer consists of an isotropic mixture of MgCO

_{3}and MgCl

_{2}. In practice, the magnesium block will be pitted and have holes that will presumably affect its corrosive properties as well [42]; this is currently being explored by the authors. There is thus plenty of scope to improve the current model, in order to describe more realistically the corrosion properties of magnesium based orthopaedic implants in vivo. Nevertheless, the current model provides a promising initial step into a theoretical understanding of magnesium corrosion, hopefully providing useful insights to help make informed decisions on the experimental direction and design of magnesium based implant materials.

## 5. Conclusions

- Mathematically modelled Mg (or Mg alloy) corrosion in aqueous environments (e.g., cell culture medium) and the corrosion products Mg(OH)
_{2}and MgCO_{3}, forming up to three discrete regions of for Mg, Mg(OH)_{2}and MgCO_{3}, the boundaries of which move in time. - The corrosion products are treated as porous media, whereby fluid (water, CO
_{2}) and solid (Mg and its compounds) phases move separately. The reactants are transported via diffusion and advection. - The model is an advection–diffusion system with multiple moving boundaries, marking the coordinate of the interfaces between the solid phase species.
- Many of the parameters are obtainable from literature, leaving three free parameters: the reaction rate between Mg (or Mg alloy) and water (k), and the solid volume fractions of Mg(OH)
_{2}(${\epsilon}_{1}$) and MgCO_{3}(${\epsilon}_{2}$) layers. - There are two key timescales for Mg corrosion process, namely that of complete corrosion of the original Mg block (${T}_{\alpha}$) and Mg(OH)
_{2}(${T}_{\beta}$), so that at ${T}_{\beta}$ all that remains is a MgCO_{3}block. Numerical solutions demonstrated that, over a wide range of parameters,- –
- ${T}_{\alpha}\ll {T}_{\beta}$, the original Mg block is short-lived relative to the complete corrosion process.
- –
- k and ${\epsilon}_{1}$ affect ${T}_{\alpha}$, whilst having a little effect on ${T}_{\beta}$. The latter is affected most by ${\epsilon}_{2}$. To substantially prolong Mg presence, Mg alloys must have the effect of reducing the reaction rate k.
- –
- geometry has an impact on the corrosion timescales.
- –
- complete corrosion is described well by the law ${T}_{\beta}\propto {S}_{0}^{2}/(1-{\epsilon}_{2})$, where ${S}_{0}$ is the original radius (or size) of the Mg block.

- Relatively simple in vitro experimentation and CT scans can be used to inform the free parameters, thereby enabling the model to predict the outcome in situations more challenging to undertake experimentally.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Change of Variables

## Appendix B. Small Time Asymptotics

#### Appendix B.1. Phase 1.1

#### Appendix B.2. Phase 2.1

#### Appendix B.3. Phase 2.2

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**Figure 1.**A physiochemical schematic of the magnesium corrosion system used in the model for cylindrical and spherical geometries. The pure magnesium or magnesium alloy exists in the core (Zone 0, $0\le r<\alpha \left(t\right)$), the magnesium hydroxide forms a middle layer (Zone 1, $\alpha \left(t\right)<r<\beta \left(t\right)$) and the outer layer consists of magnesium carbonate (Zone 2, $\beta \left(t\right)<r<S\left(t\right)$).

**Figure 2.**Plots of the dimensionsionless variables $\alpha ,\beta $ and S against t in cylindrical geometry using ${\epsilon}_{1}=0.6$, ${\epsilon}_{2}=0.4$, $\kappa =0.04$, the parameters in Table 4 and ${S}_{0}=1$. The dashed lines show $t={T}_{\alpha =\beta}$ (

**left**), $t={T}_{\alpha}$ (

**middle**) and ${T}_{\beta}$ (

**right**).

**Figure 3.**Plots of the dimensionsionless variables concentrations ${W}_{1},{W}_{2}$ (left) and ${C}_{2}$ (right) at, from top to bottom, the start of Phases 2.1 ($t={\tau}_{0}={10}^{-8}$), end of Phase 2.1 ($t={T}_{\alpha =\beta}\approx 0.095$), Phase 2.2 ($t={T}_{\alpha =\beta}+{\tau}_{1}$, with ${\tau}_{1}=0.356$, see Appendix B.3), start of Phases 2.3 ($t={T}_{\alpha}\approx 33.1$) and the end of 2.3 ($t={T}_{\beta}\approx 1291$) in cylindrical geometry. In the left-hand panel, the solid lines are ${W}_{1}$ and the dashed lines ${W}_{2}$. The vertical dotted lines indicate from right to left, $r=S$, $r=\beta $ (top 3 plots) and $r=\beta $ (top 2 plots). The parameters are ${\epsilon}_{1}=0.6$, ${\epsilon}_{2}=0.4$, $\kappa \approx 0.04$, ${S}_{0}=1$ and the rest listed in Table 4.

**Figure 4.**Plots of the dimensionsionless variables $\alpha ,\beta $ and S against t, from left to right, Cartesian, cylindrical and spherical geometry using ${\epsilon}_{1}=0.6$, ${\epsilon}_{2}=0.4$, $\kappa \to \infty $, the parameters in Table 4 and ${S}_{0}=1$. The dashed lines show $t={T}_{\alpha}$ (

**left**) and ${T}_{\beta}$ (

**right**).

**Figure 5.**Plots of the dimensionsionless ${T}_{\alpha}$ and ${T}_{\beta}$ against the initial magnesium block size, ${S}_{0}$, for each of the principle geometries. The parameters used are ${\epsilon}_{1}=0.6$, ${\epsilon}_{2}=0.4$, $\kappa \approx 0.3$ ($k=0.5\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{4}/\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\mathrm{day}$) and parameters in Table 4.

**Figure 6.**Plots of the dimensionsionless ${T}_{\alpha}$ and ${T}_{\beta}$ against ${\epsilon}_{1}$ (

**left**) and ${\epsilon}_{2}$ (

**right**) for $\kappa =0.03,6$ and $\kappa \to \infty $, with the remaining parameters listed in Table 4 and ${S}_{0}=1$.

**Figure 7.**Contour map of the dimensionsionless ${T}_{\alpha}$ for $\kappa $ with ${\epsilon}_{1}=0.4$ (

**top**) and $\kappa $ against ${\epsilon}_{2}$ with ${\epsilon}_{1}=0.6$ (

**bottom**) in spherical geometry. The remaining parameters are listed in Table 4 and ${S}_{0}=1$.

**Figure 8.**Plots of the dimensionsionless ${T}_{\alpha}$ against $\kappa $ for the three principle geometries, with ${\epsilon}_{1}=0.6$${\epsilon}_{2}=0.4$, ${S}_{0}=1$ and the parameters listed in Table 4.

**Figure 9.**Plots of the evolution of the dimensionsionless variables $\alpha ,\beta $ and S resulting from three choices of fluid phase advection velocity, ${V}_{i}$, namely ${V}_{i}={v}_{{f}_{i}}$ (as proposed in the current model, solid lines), ${V}_{i}=0$ (dotted) and ${V}_{i}={v}_{{s}_{i}}$ (as used in [16], dashed). The left plot shows the full evolution of the interfaces and the right plot shows the results around $t={T}_{\alpha}$. The results are using cylindrical geometry, with ${\epsilon}_{1}=0.6$, ${\epsilon}_{2}=0.4$ and parameters in Table 4 and ${S}_{0}=1$.

**Table 1.**Notation used in the model, where ${\rho}_{i}={M}_{i}{\mu}_{i},{\omega}_{\alpha}={\mu}_{0}/{\mu}_{1}$ and ${\omega}_{\beta}={\mu}_{1}/{\mu}_{2}$.

Name | Notation |
---|---|

Zone | i |

Solid fraction | ${\epsilon}_{i}$ |

Mass density | ${\rho}_{i}$ |

Mass/Mol | ${M}_{i}$ |

Mol/Volume | ${\mu}_{i}$ |

Solid velocity | ${v}_{{s}_{i}}$ |

Fluid velocity | ${v}_{{f}_{i}}$ |

**Table 2.**Relevant boundary conditions for each of the Mg corrosion phases for the two cases $k\to \infty $ and $k<\infty $.

Phase | Time | b.c.s | ${\mathit{v}}_{{\mathit{s}}_{\mathit{i}}}$ and ${\mathit{v}}_{{\mathit{f}}_{\mathit{i}}}$ | |
---|---|---|---|---|

1.1 | $k\to \infty $ | $0\le t\le {T}_{\alpha}$ | (13), (15) | (17) |

1.2 | ${T}_{\alpha}<t\le {T}_{\beta}$ | (14), (15) | (17) | |

2.1 | $k<\infty $ | $0\le t\le {T}_{\alpha =\beta}$ | (16) | (19) |

2.2 | ${T}_{\alpha =\beta}<t\le {T}_{\alpha}$ | (13), (15) | (17) | |

2.3 | ${T}_{\alpha}<t\le {T}_{\beta}$ | (14), (15) | (18) |

**Table 3.**List of model variables, their interpretation and, where possible, estimated values from the literature. † unknown parameters in the model. “S” indicates standard textbook references and “D” derived from formula in Table 1.

Parameter | Value | Units | Description | Source |
---|---|---|---|---|

${D}_{W}$ | $2.85$ | ${\mathrm{cm}}^{2}/\mathrm{day}$ | Diffusion coefficient of H_{2}O | [29] |

${D}_{C}$ | $1.66$ | ${\mathrm{cm}}^{2}/\mathrm{day}$ | Diffusion coefficient of CO_{2} | [30] |

${M}_{0}$ | $24.3$ | g/mol | Molecular mass of Mg | S |

${M}_{1}$ | $58.3$ | g/mol | Molecular mass of Mg(OH)_{2} | S |

${M}_{2}$ | $84.3$ | g/mol | Molecular mass of MgCO_{3} | S |

${M}_{W}$ | 18 | g/mol | Molecular mass of H_{2}O | S |

${M}_{C}$ | 44 | g/mol | Molecular mass of CO_{2} | S |

${\rho}_{0}$ | $1.74$ | $\mathrm{g}/{\mathrm{cm}}^{3}$ | Mass density of Mg | S and [31] |

${\rho}_{1}$ | $2.34$ | $\mathrm{g}/{\mathrm{cm}}^{3}$ | Mass density of Mg(OH)_{2} | S |

${\rho}_{2}$ | $2.96$ | $\mathrm{g}/{\mathrm{cm}}^{3}$ | Mass density of MgCO_{3} | S |

${W}_{0}^{*}$ | 1 | $\mathrm{g}/{\mathrm{cm}}^{3}$ | Concentration of H_{2}O in human body | S |

${C}_{0}^{*}$ | 0.0011 | $\mathrm{g}/{\mathrm{cm}}^{3}$ | Concentration of CO_{2} in human body | [32] |

${w}_{\alpha}$ | $1.8$ | - | Molar density ratio of Mg and Mg(OH)_{2} | D |

${w}_{\beta}$ | $1.1$ | - | Molar density ratio of Mg(OH)_{2} and MgCO_{3} | D |

${\epsilon}_{1}$ | † | - | Fraction of magnesium hydroxide | - |

${\epsilon}_{2}$ | † | - | Fraction of magnesium carbonate | - |

k | † | ${\mathrm{cm}}^{4}/\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\mathrm{day}$ | Rate of reaction between Mg and H_{2}O | - |

**Table 4.**List of dimensionless parameter values calculated from the values listed in Table 3 and Equation (22); † being the free parameters.

Parameter | Value |
---|---|

${D}_{W}$ | $1.5625$ |

${\gamma}_{0}$ | $1.2889$ |

${\gamma}_{1}$ | $0.7225$ |

${\gamma}_{2}$ | $1605.5$ |

${w}_{\alpha}$ | $1.8$ |

${w}_{\beta}$ | $1.1$ |

${\epsilon}_{1}$ | † |

${\epsilon}_{2}$ | † |

$\kappa $ | † |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ahmed, S.K.; Ward, J.P.; Liu, Y.
Numerical Modelling of Effects of Biphasic Layers of Corrosion Products to the Degradation of Magnesium Metal In Vitro. *Materials* **2018**, *11*, 1.
https://doi.org/10.3390/ma11010001

**AMA Style**

Ahmed SK, Ward JP, Liu Y.
Numerical Modelling of Effects of Biphasic Layers of Corrosion Products to the Degradation of Magnesium Metal In Vitro. *Materials*. 2018; 11(1):1.
https://doi.org/10.3390/ma11010001

**Chicago/Turabian Style**

Ahmed, Safia K., John P. Ward, and Yang Liu.
2018. "Numerical Modelling of Effects of Biphasic Layers of Corrosion Products to the Degradation of Magnesium Metal In Vitro" *Materials* 11, no. 1: 1.
https://doi.org/10.3390/ma11010001