# A perspective on effective medium models of thermal conductivity in (ultra)nanocrystalline diamond films

## TL;DR

This paper compares different effective medium theory models for thermal conductivity in nanocrystalline diamond films, highlighting the impact of grain size, boundary resistance, and intra-grain scattering, and introduces a more consistent inclusion shape approximation.

## Contribution

It introduces an EMT model with arbitrarily shaped inclusions, specifically cubic, and compares it with spherical models to improve consistency in thermal conductivity analysis.

## Key findings

- Good agreement between cubic and spherical EMT models due to geometrical considerations.
- Thermal conductivity is significantly influenced by intra-grain scattering and boundary Kapitza resistance.
- Kapitza resistance effects are minimal due to geometrical factors, especially at small grain sizes.

## Abstract

Thermal conductivity of nanocrystalline and ultra-nanocrystalline films is analyzed with effective medium theory (EMT) models. The existing EMT models use the spherical inclusion approximation. Although this approximation works quite well it is inconsistent, mostly with respect to the maximal packing of 74{\%}, which may be unrealistic for polycrystalline films. To check the consistency of these models we devise an EMT model with arbitrarily shaped inclusions. We pick the EMT model with cubic inclusions and we compare its results with the results of the EMT model with spherical inclusions. It is found a very good agreement between both calculations. This agreement is explained by general geometrical arguments. We further employ these models to analyze thermal conductivity of nanocrystalline and ultra-nanocrystalline diamond films. It is noticed that the effective conductivity is strongly affected not only by the boundary Kapitza resistance but also by intra-grain scattering for grain sizes below 100 nm. Generally, both intra-grain conductivity and Kapitza resistance increase with grain size. However, the effect of Kapitza resistance increase is negligible due to the geometrical factor accompanying Kapitza resistance contribution to the effective conductivity.