Grain shape and texture effect on electrical characterization of semi-conductor semi-insulator mixture

Abstract

Natural rocks are available in different ranges of shapes, sizes, and orientations. The orientation and structure of grains in rocks have a big influence on the electrical behavior of the rock, especially when semi-insulator grains cut the path of electrical links between electrodes. The grain shape effect on electrical characteristics was demonstrated using a mixture of a relatively semi-conducting phase (hematite) and a relatively semi-insulating phase (sand), i.e., hematitic sandstone. A hematitic sandstone mixture gives an abnormally high dielectric permittivity and conductivity behavior. Mixture laws are unable to justify this abnormal behavior. On the other hand, the effective medium theory expresses an unusual approach to this issue. The electrical characteristics of the measured samples were likely to be seriously influenced by the grain shape of the constituents. The effect of grain shape may be greater than the effect of hematite concentrations as a semi-conductor. The successful medium hypothesis and experimental evidence have a high level of agreement. It was possible to cover the experimental spectrum of varieties (for dielectric permittivity and conductivity) at different concentrations, within the framework of grain shape varieties used here, using the effective medium hypothesis. The numerical model used agrees well with the experimental results.

Appendix. Effective medium hypothesis for spheroidal grains

The effective medium equation for a random mixture consisting of phases α with complex dielectric permittivities ε_α and spheroidal shaped grains is given by

$$\frac{1}{3}\sum_{\alpha }\sum_{i=1}^{3}\frac{{\varepsilon }_{\alpha }-{\varepsilon }}{{L}_{\alpha i}{\varepsilon }_{\alpha }+(1-{L}_{\alpha i}){\varepsilon }}{p}_{\alpha }=0$$

(3)

This formula takes into consideration the random orientations of the grains in the x, y, and z directions $\left(i=\text{ 123}\right)$, and ${L}_{\alpha i}$ is the depolarization factor of ${\alpha }^{th}$ the constituent in the ${i}^{th}$ direction.

The values ${L}_{\alpha i}$ are determined by the eccentricity e of the spheroid. For prolate spheroids (~ needles), with semi-axes ${\text{b}}_{1}>{\text{b}}_{2}={\text{b}}_{3}$ (Mendelson and Cohen 1982) and $e={\left[1-{\left({~}^{{b}_{2}}\!\left/ \!\!{~}_{{b}_{1}}\right.\right)}^{2}\right]}^{{~}^{1}\!\left/ \!\!{~}_{2}\right.}$, the depolarization factor in the direction of the spheroid axis of symmetry is

$${L}_{1}=\frac{1-{e}^{2}}{2{e}^{3}}\left(\text{ln}\frac{1+e}{1-e}-2e\right)$$

(4)

For oblate spheroids (~ discs), with ${\text{b}}_{1}<{\text{b}}_{2}={\text{b}}_{3}$, $e={\left[{\left({~}^{{b}_{2}}\!\left/ \!\!{~}_{{b}_{1}}\right.\right)}^{2}-1\right]}^{{~}^{1}\!\left/ \!\!{~}_{2}\right.}$ and

$${L}_{1}=\frac{1+{e}^{2}}{{e}^{3}}\left(e-{\tan }^{-1}e\right)$$

(5)

With ${L}_{1}+{L}_{2}+{L}_{3}=1$,${L}_{2}={L}_{3}$.

For spherical grains ${L}_{1}={L}_{2}={L}_{3}={~}^{1}\!\left/ \!\!{~}_{3}\right.$.

For needles ${L}_{1}\to 1,{L}_{2}={L}_{3}\approx 0$.

For discs ${L}_{1}\to 0,{L}_{2}={L}_{3}\approx {~}^{1}\!\left/ \!\!{~}_{2}\right.$.

Equation (3) is an algebraic equation of the second-order $\varepsilon$ when the two media are spheres. For spheroids or more than two phases the order of the algebraic equation increases. To solve such a nonlinear equation to obtain $\varepsilon$, M $u$ ller’s method is used (Conte and De Boor 1980) (Fig. 3). This method is similar to the secant method in which we determine, from the approximations x_i, x_i-1 for a root of $f\left(x\right)=0$, the next approximation ${x}_{i+1}$ as the zero of the linear polynomial $p\left(x\right)$ which goes through the two points $\left\{{x}_{i},f\left({x}_{i}\right)\right\}$ and$\left\{{x}_{i-1},f\left({x}_{i-1}\right)\right\}$. In M $u$ ller’s method, the next approximation ${x}_{i+1}$ is found as a zero of the parabola defined by$p\left(x\right)$, which goes through the three points $\left\{{x}_{i},f\left({x}_{i}\right)\right\}$, $\left\{{x}_{i-1},f\left({x}_{i-1}\right)\right\}$, and$\left\{{x}_{i-2},f\left({x}_{i-2}\right)\right\}$,

$$p\left(x\right)=f\left({x}_{i}\right)+f\left[{x}_{i},{x}_{i-1}\right]\left(x-{x}_{i}\right)+f\left[{x}_{i},{x}_{i-1},{x}_{i-2}\right]\left(x-{x}_{i}\right)\left(x-{x}_{i-1}\right)$$

(6)

where

$$f\left[{x}_{i},{x}_{i-1}\right]=\left(f\left({x}_{i}\right)-f\left({x}_{i-1}\right)\right)/{h}_{i},$$

(7)

$${h}_{i}={x}_{i}-{x}_{i-1}$$

(8)

$$f\left[{x}_{i},{x}_{i-1},{x}_{i-2}\right]=\left(f\left[{x}_{i},{x}_{i-1}\right]-f\left[{x}_{i-1},{x}_{i-2}\right]\right)/\left({h}_{i}+{h}_{i-1}\right)$$

(9)

The three points are used as approximations for a root of $f\left(x\right)$, and a zero of the equation of the parabola $p\left(x\right)$ gives an improved solution (Fig. 4).

The function $p\left(x\right)$ can be written in the form

$$p\left(x\right)=f\left({x}_{i}\right)+\left(x-{x}_{i}\right){c}_{i}+f\left[{x}_{i},{x}_{i-1},{x}_{i-2}\right]{\left(x-{x}_{i}\right)}^{2}$$

(10)

with

$${c}_{i}=f\left[{x}_{i},{x}_{i-1}\right]+f\left[{x}_{i},{x}_{i-1},{x}_{i-2}\right]\left({x}_{i}-{x}_{i-1}\right)$$

(11)

A zero ${\alpha }_{a}$ of the parabola $p\left(x\right)$ satisfies (Conte and De Boor, 1980)

$${\alpha }_{a}-{x}_{i}=\frac{-2f\left({x}_{i}\right)}{{c}_{i}\pm {\left\{{c}_{i}^{2}-4f\left({x}_{i}\right)f\left[{x}_{i},{x}_{i-1},{x}_{i-2}\right]\right\}}^{1/2}}$$

(12)

If we choose the sign in Eq. (12) so that the denominator will be as large in magnitude as possible and if we label ${\alpha }_{a}-{x}_{i}={h}_{i+1}$, then the next approximation to a zero of $f\left(x\right)$ is taken to be

$${x}_{i+1}={x}_{i}+{h}_{i+1}$$

(13)

The process is then repeated using ${x}_{i-1}$, ${x}_{i}$, and ${x}_{i+1}$ as the three basic approximations.

This approach is iterative, does not require evaluating the function’s derivative, obtains both real and complex roots, and does not necessitate a precise starting solution (Conte and De Boor 1980).

Starting with a unit volume of the insulator (or semi-conductor) for which the exact solution is known at any frequency, one may obtain a starting solution for the appropriate root (among the various roots of the equation). The volume conductor’s fraction is increased by a slight amount, and the previous concentration’s solution is used as a starting solution. Since the improvements in dielectric permittivity and conductivity are so abrupt near the percolation threshold, the concentration increment is considered to be very tiny. Since the adjustments in dielectric permittivity and conductivity are slight away from the percolation threshold, a greater increment is used. To obtain the dielectric permittivity at any concentration value, the process is iterated.

A comparison with Tobochnik et al. (1990) at direct current for a model with a large contrast in the conductivities of the constituents was made as a test for the solution (as in hematite $\sigma ={10}^{-2}$ and sand $\sigma ={10}^{-10}$). Tobochnik’s corresponding conductor–superconductor results, which hold at relatively low conductor concentrations, and Tobochnik’s corresponding semi-conductor–semi-conductor results, which hold at relatively high semi-conductor concentrations, agree with the computed results.

The well-known formula (Dukhin 1971) was used to compare the mixture law to the effective medium theories:

$$\varepsilon ={\varepsilon }_{1}\frac{2{\varepsilon }_{1}+{\varepsilon }_{2}-2{f}_{2}({\varepsilon }_{1}-{\varepsilon }_{2})}{2{\varepsilon }_{1}+{\varepsilon }_{2}+{f}_{2}({\varepsilon }_{1}-{\varepsilon }_{2})}$$

(14)

where $({\varepsilon }_{1})$ is the complex conductivity of the dispersive medium, (ε₂) is the complex conductivity of the dispersed phase, f₂ is the volume fraction of the dispersed phase, and (f₂ << 1).