A Mathematical Model for Simulation of Intergranular μ-Capacitance as a Function of Neck Growth in Ceramic Sintering

Abstract

In this paper we will define a new mathematical model for predicting an evolution of an equivalent intergranular μ-capacitance during ceramic sintering. The contact between two adjacent grains will be defined as a structure that forms a μ-capacitor recognized as an intergranular μ-capacitor unit. It will be assumed that its μ-capacitance changes as the neck grows by diffusion. Diffusion mechanisms responsible for transport matter from the grain boundary to the neck are the volume diffusion and grain boundary diffusion. Such model does not need special geometric assumptions because the microstructural development can be simulated by a set of simple local rules and overall neck growth law which can be arbitrarily chosen. To find the total capacitance we will identify μ-capacitors in series and in parallel. More complicated connections of μ-capacitors will be transformed into simpler structure using delta to star transformation and/or star to delta transformation. In this way some μ-capacitors will be step-by-step replaced by their equivalent μ-capacitors. The developed model can be applied for the prediction of an evolution of the intergranular capacitance during ceramic sintering of BaTiO₃ system with spherical particle distributions.

Appendices

Appendix 1: Neck Radius Computation

In computation of the time-dependent neck radius we will use the densification model of sintering based on Coble’s two-grain model [15] and the extended model [16] in which diffusion mechanisms responsible for transport matter from the grain boundary to the neck are the volume diffusion and grain boundary diffusion (Figure 1a). Next assumptions will be made: volume conservation, center-to-center approach, and neck geometry simplified by straight line. For simulation purposes increasing the grain radius, induced by overlapping (defined by radius X ₁) and maintaining the volume conservation of the two-grain model, will be neglected. All mentioned geometrical simplifications and notations are shown in Figure 12: the volume of the spherical cap

$$\displaystyle \begin{aligned}V_{\text{SphCap}}=\dfrac{\pi h}{6} (3x_1^2+h^2)\ \ \ \left(h=R-\dfrac D2\right) ,\end{aligned}$$

the volume of the ring

$$\displaystyle \begin{aligned}V_{\text{Ring}}=V_{\text{Cyl}}-V_{\text{SphSeg}} ,\end{aligned}$$

where the volume of the cylinder is defined as

$$\displaystyle \begin{aligned}V_{\text{Cyl}}=\pi x_2^2y_2 ,\end{aligned}$$

and the volume of the spherical segment as

$$\displaystyle \begin{aligned}V_{\text{SphSeg}}=\dfrac16 \pi y_2 (3x_1^2+3x_2^2+y_2^2)\ .\end{aligned}$$

Taking into account distributing intersected volume to the neck width (at constant grain volume), the next algorithm (for numerical approaching, where 𝜖 is small enough positive number) will be applied for computation of the neck radius X ₂:

$$\displaystyle \begin{aligned} \begin{array}{l} \quad \quad \quad \quad x_2 \mapsto x_1\\ \quad \quad \quad \quad \mathbf{{While}}\ \left|V_{\text{Ring}}-V_{\text{SphCap}}\right| > \epsilon \\ \quad \quad \quad \quad \quad \quad x_2 \mapsto x_2+\varDelta x\\ \quad \quad \quad \quad \mathbf{{Wend}}\end{array} \end{aligned}$$

Fig. 14

Four-grain model as a representation of μ-capacitors in star connection. (a) 3D model and (b) appropriate electrical scheme

Appendix 2: Delta to Star Transformation

Structure in Figure 13b can be transformed to the structure in Figure 14b as follows [17]:

$$\displaystyle \begin{aligned} \mu C_1=\dfrac{\mu C_{12}\cdot\mu C_{13}+\mu C_{12}\cdot\mu C_{23}+\mu C_{13}\cdot\mu C_{23}}{\mu C_{23}}\ , {} \end{aligned} $$

(9)

$$\displaystyle \begin{aligned} \mu C_2=\dfrac{\mu C_{12}\cdot\mu C_{13}+\mu C_{12}\cdot\mu C_{23}+\mu C_{13}\cdot\mu C_{23}}{\mu C_{13}}\ , {} \end{aligned} $$

(10)

$$\displaystyle \begin{aligned} \mu C_3=\dfrac{\mu C_{12}\cdot\mu C_{13}+\mu C_{12}\cdot\mu C_{23}+\mu C_{13}\cdot\mu C_{23}}{\mu C_{12}}\ \ .{} \end{aligned} $$

(11)

Appendix 3: Star to Delta Transformation

Structure in Figure 14b can be transformed to the structure in Figure 13b as follows [17]:

$$\displaystyle \begin{aligned} \mu C_{12}=\dfrac{\mu C_1\cdot\mu C_2}{\mu C_1+\mu C_2+\mu C_3}\ , {} \end{aligned} $$

(12)

$$\displaystyle \begin{aligned} \mu C_{13}=\dfrac{\mu C_1\cdot\mu C_3}{\mu C_1+\mu C_2+\mu C_3}\ , {} \end{aligned} $$

(13)

$$\displaystyle \begin{aligned} \mu C_{23}=\dfrac{\mu C_2\cdot\mu C_3}{\mu C_1+\mu C_2+\mu C_3}\ \ .{} \end{aligned} $$

(14)