Mesoscale Thermal Conductivity and Resistivity of Polycrystalline Graphite

Abstract

With rapid advances in Microelectromechanical systems (MEMS) and Nanotechnology applications, it has become essential to predict the properties of materials at length scales that are significantly below the macroscale. Understanding the structure-property relationship at various length scales is vital in the development of new materials for various critical applications. This thesis focuses on estimating the effective thermal conductivity of polycrystalline graphite at micro, meso and macroscales. At the length scale of a single crystal, graphite generally has an anisotropic and uniaxial thermal character in which two of its principal conductivities are equal and different from the third principal conductivity. At intermediate mesoscales, an aggregate of graphite single crystals collectively exhibits an anisotropic thermal response with six independent components for each of the mesoscale conductivity and resistivity tensors, and thereby possesses three independent thermal conductivity invariants. With random orientation distribution and upon ensemble averaging, an isotropic conductivity is obtained. This research examines the anisotropic heat conduction in random polycrystalline graphite under uniform essential (Dirichlet) and uniform natural (Neumann) boundary conditions. Rigorous bounds are obtained on the apparent thermal conductivity of polycrystalline graphite by numerically solving over 2700 boundary value problems. All the six independent components of the conductivity (and resistivity) tensor are obtained in addition to the tensor invariants by solving the resulting eigenvalue problem. The bounds obtained are compared to the well-known results proposed by Voigt, Reuss and the upper/lower Hashin-Shtrikman bounds. It is concluded that bounds obtained in this study show an improvement of 53 % when compared to the Hashin-Shtrikman bounds. The scale-dependent statistics for the invariants of thermal conductivity and resistivity tensors are also obtained including mean, standard deviation, coefficient of variation, skewness and kurtosis. Several distributions are used to construct the statistical fits (Beta, Gumbel Max, Chi-squared Weibull and Rayleigh distribution) and the goodness of fit is evaluated using the Kolmogorov-Smirnov, Anderson-Darling and Chi-Squared techniques. Test results conclude that the Beta probability distribution captures the statistics of the tensor invariants better than other distributions.