The thermal conductivity of particulate composites by the use of a polyhedral model

Abstract

In this paper, the authors introduce a body-centered polyhedral model capable of simulating the periodic structure of particulate composites, in order to estimate the thermal conductivity of a general class of this type of materials. This model takes into account the arrangement of internal and neighboring particles in the form of three distinct deterministic configurations, along with the concept of interphase on the thermal and mechanical properties of the composite, which is assumed to be macroscopically homogeneous and isotropic. Next, by means of this advanced multilayer model, an explicit analytical expression to evaluate the thermal conductivity of this type of composites is derived. The theoretical predictions were compared with the experimental results found in the literature, as well as with the theoretical values yielded by some reliable formulae derived from other workers, and a reasonable agreement was observed.

Appendix

Let us present some formulae for the thermal conductivity of particulate composites found in the literature.

The Maxwell–Eucken formula [21]

$$ {K}_c={K}_m\left(\frac{K_f+2{K}_m+2{U}_f\left({K}_f-{K}_m\right)}{K_f+2{K}_m+-\left({K}_f-{K}_m\right)}\right) $$

(41)

The Bruggeman formula [22]

$$ 1-{U}_f=\frac{K_f-{K}_c}{K_f-{K}_m}{\left(\frac{K_m}{K_c}\right)}^{1/3} $$

(42)

This implicit expression can be equivalently modified to a cubic equation by setting.

$x=\sqrt[3]{K_c}$ with $x\in {R}_{+}^{\ast }$.

Thus, it implies that

$$ \left(1-{U}_f\right)\frac{x}{\sqrt[3]{K_m}}=\frac{K_f-{x}^3}{K_f-{K}_m}\iff \frac{\left(1-{U}_f\right)\cdot \left({K}_f-{K}_m\right)}{\sqrt[3]{K_m}}x={K}_f-{x}^3\iff $$

$$ {x}^3+\frac{\left(1-{U}_f\right)\cdot \left({K}_f-{K}_m\right)}{\sqrt[3]{K_m}}x-{K}_f=0 $$

This cubic equation can be solved with respect to the auxiliary variable x using the well-known Cardano’s technique provided that the variable U _f takes the distinct values 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.65.

Next, the corresponding values of Κ _c can be easily calculated.

The Lewis–Nielsen formula [23]

$$ {K}_c={K}_m\frac{1+A\cdot B\cdot {U}_f}{1-B\cdot \psi \cdot {U}_f} $$

(43)

with

$$ B=\frac{K_f/{K}_m-1}{K_f/{K}_m+A};\psi =1+\frac{K_f/{K}_m-1}{K_f/{K}_m+A}{U}_f $$

Also, A is the shape coefficient for the filler particles which, for spherical ones, is equal to 1.5.

The Kytopoulos–Sideridis formula [28]

$$ \frac{1}{\kern0.24em {K}_c}=\frac{1}{K_m}+\frac{\left({K}_m-{K}_f\right){U}_f}{K_f{K}_m}\max \left\{{k}_1,{k}_2\right\} $$

(44)

where k ₁ and k ₂ are the dimensionless parameters somewhat analogous to the packing factor for periodic particulate composites defined by Theocaris in Ref. [36]. Evidently, these parameters lie between 0 and 1.

The Venetis–Sideridis formula [31]

$$ {K}_c=\frac{K_m{K}_f{\overline{K}}_i}{K_f{\overline{K}}_i\left(\frac{r_1^3}{r_5^3}+\frac{r_5^3-{r}_4^3}{r_5^3}\right)+{K}_m{\overline{K}}_i\left(\frac{r_3^3-{r}_2^3}{r_5^3}\right)+{K}_f{K}_m\left(\frac{r_2^3-{r}_1^3}{r_5^3}+\frac{r_4^3-{r}_3^3}{r_5^3}\right)} $$

(45)

where r ₁ to r ₅ denote the radii of a coaxial five-phase spherical model arising from the transformation of the non-body-centered cubic RVE.