A schematic representation of a fiber bundle and important dimensions for the model are shown in Figure 1. The number of fibers is *n* (-); the initial radius of the fibers is *r*_{0} (m); the length of fibers is *l* (m).

Figure 1 Left: Schematic representation of a fiber bundle and geometrical dimensions. Right: micrograph of a fiber bundle.

The model involves the following assumptions. As a simplification, this model is deterministic and all fibers are assumed to have the same initial radius, which is *r*_{0}; and the cross-sectional surface area at the end of the fibres is assumed to be negligible in calculations of the surface area. The length of the long fibers *l* is assumed to be constant during the whole dissolution process. During the whole degradation process, the density of the glass material stays constant $\left({\rho}_{glass}\right)\text{\hspace{0.17em}}.$

Dissolution is a surface reaction. In a general case, the rate of the dissolution is dependent on the constant describing the rate of the reaction (*K*_{0}), the glass surface area exposed to water (*S*), the availability of water $\left({C}_{{H}_{2}O}\right)$ and the order of the reaction $\left({n}_{reaction}\right).$The global model (general case) can then be mathematically expressed as Equation 4.

$$\frac{\partial m}{\partial t}={K}_{0}S{C}_{{H}_{2}O}^{{n}_{reaction}}$$(4)In case of infinite availability of water, the rate of reaction becomes independent of the reactant (water) concentration, and the reaction order *n*_{reaction} becomes 0. In infinite water availability conditions, the surface reaction can be well-described with zero-order kinetics [16, 24], which can then be represented by a differential Equation 5:

$$\frac{\partial m}{\partial t}={K}_{0}S$$(5)where *m*(g) is a total cumulative mass dissolved after time *t* (s), *K*_{0} (g/m^{2}∙s) is a zero-order reaction kinetic constant and *S* (m^{2}) is the glass surface area in contact with water.

As the reaction proceeds, the radius of the fibers is reduced and the total surface area (*S*) is decreased, thus leading to a decrease in the rate of mass loss as seen in Equation 5. The overall ion release rate decreases proportionally to the decrease in total surface area or a decrease in fiber radius.

$$\frac{S\left(t\right)}{{S}_{0}}=\frac{2\pi nlr\left(t\right)}{2\pi nl{r}_{0}}=\frac{r\left(t\right)}{{r}_{0}}$$(6)It can be seen that $\frac{S\left(t\right)}{{S}_{0}}=\frac{r\left(t\right)}{{r}_{0}}\text{\hspace{0.17em}}.$Thus, the ion release rate decelerates linearly with a decrease in fiber radius. The mass loss has to slow down as the available surface area is reduced. Thus, the cumulative mass loss kinetic curve deviates from a linear dissolution.

The volume of a single fiber is *πr*^{2} l, where *l* is the cylinder length and *r* is the cylinder radius. For *n* fibers, the volume is *πr*^{2} l and mass is ${\rho}_{glass}n\pi {r}^{2}l\text{\hspace{0.17em}}.$The surface area of a single fiber is *2nπrl*. For *n* fibers it is *2nπrl*. Substituting mass and surface area expressed in such terms into Equation 5,

$$\frac{\partial {\rho}_{glass}n\pi {r}^{2}l}{\partial t}={K}_{0}\cdot 2n\pi rl$$(7)$$\frac{\partial {r}^{2}}{\partial t}=\frac{2{K}_{0}}{{\rho}_{glass}}r$$(8)Substituting then *r*^{2} with the following *z* = *r*^{2},

$$\frac{\partial z}{\partial t}=\frac{2{K}_{0}}{{\rho}_{glass}}\sqrt{z}$$(9)where $\frac{2{K}_{0}}{{\rho}_{glass}}$is a constant. Dividing both sides by $\sqrt{z},$

$$\frac{\frac{\partial z}{\partial t}}{\sqrt{z}}=\frac{2{K}_{0}}{{\rho}_{glass}}$$(10)Both sides are then integrated with respect to t,

$$\int \frac{\frac{\partial z}{\partial t}}{\sqrt{z}}dt=}{\displaystyle \int \frac{2{K}_{0}}{{\rho}_{glass}}dt$$(11)Integrating and solving for $\sqrt{z},$

$$\sqrt{z}=\frac{1}{2}\left(\frac{2{K}_{0}}{{\rho}_{glass}}t+{c}^{\prime}\right)$$(12)$$\sqrt{z}=\frac{{K}_{0}}{{\rho}_{glass}}t+{{c}^{\prime}}^{\prime}$$(13)where *c’* and *c’’* are arbitrary constants after integration. Substituting *r*^{2} back into the equation:

$$\sqrt{{r}^{2}}=\frac{{K}_{0}}{{\rho}_{glass}}t+{{c}^{\prime}}^{\prime}$$(14)$$\pm r=\frac{{K}_{0}}{{\rho}_{glass}}t+{{c}^{\prime}}^{\prime}$$(15)It can be seen that the radius reduction is linear with time, where the proportionality is given by the zero-order kinetic constant *K*_{0} (g/m^{2}∙s) and the density of the glass (g/m^{3}). A linear radius reduction of fibers with time was previously experimentally observed in another work [16]. Since the radius reduction depends on the initial radius *r*_{0} of the fibers (*m*), the arbitrary constant *c’’* (*m*) is equal to the initial radius.

The kinetic model equation for fiber radius reduction then becomes as shown in Equation 16:

$$r={r}_{0}-\frac{{K}_{0}}{{\rho}_{glass}}t$$(16)Returning to the mass loss kinetics (Equation 5),

$$\frac{\partial m}{\partial t}={K}_{0}S={K}_{0}\cdot 2n\pi rl$$(5)Substituting *r* for the radius reduction kinetic Equation 28, the final mass loss kinetic model equation in differential form is obtained (Equation 17):

$$\frac{\partial m}{\partial t}=2n\pi l\left({r}_{0}{K}_{0}-\frac{{K}_{0}^{2}}{{\rho}_{glass}}t\right)$$(17)Integrating the obtained Equation 17 over time *t*, the integral model equation is obtained, describing the cumulative mass loss (ion release):

$${m}_{dissolved}={\displaystyle {\int}_{0}^{t}2n\pi l\left({r}_{0}{K}_{0}-\frac{{K}_{0}^{2}}{{\rho}_{glass}}t\right)}\text{\hspace{0.17em}}dt$$(18)Definite integral solution is then the following (Equation 19):

$${m}_{dissolved}=-\frac{1}{2}\cdot 2n\pi l\cdot t\left(\frac{{K}_{0}^{2}}{{\rho}_{glass}}t-2{r}_{0}{K}_{0}\right)$$(19)The final mass loss model kinetic equation in the integral form is then the following (Equation 20):

$${m}_{dissolved}=n\pi l\left(2{r}_{0}{K}_{0}t-\frac{{K}_{0}^{2}}{{\rho}_{glass}}{t}^{2}\right)$$(20)The solution of Equation 20 is then checked using an alternative approach. Based on the mass conservation principle, mass loss in integral form can be also written as Equation 21:

$${m}_{dissolved}={m}_{0}-m={\rho}_{glass}n\pi l\left({r}_{0}^{2}-{r}^{2}\right)$$(21)Combining Equations 16 and 21,

$$\begin{array}{l}{m}_{dissolved}={\rho}_{glass}n\pi l\left({r}_{0}^{2}-{\left({r}_{0}-\frac{{K}_{0}}{{\rho}_{glass}}t\right)}^{2}\right)=\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=n\pi l\left(2{r}_{0}{K}_{0}t-\frac{{K}_{0}^{2}}{{\rho}_{glass}}{t}^{2}\right)\end{array}$$(22)The obtained Equation 22 is the same mass loss kinetic equation in the integral form as Equation 20. The use of two alternative mathematical ways to obtain Equations 20 and 22 by the integration and by the mass conservation principle, respectively, has proven the mathematical consistency of the model. The fact that the model is adequate and physical was demonstrated by its ability to explain the experimentally observed dissolution phenomena as described later.

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