A hidden Markov model for describing turbostratic disorder applied to carbon blacks and graphene

TL;DR

This paper introduces a hidden Markov model framework to mathematically describe turbostratic disorder in layered materials like carbon blacks and graphene, enabling analysis of their scattering properties.

Contribution

It develops a novel stochastic model using hidden Markov processes and functional analysis to represent layer disorder and compute scattering cross sections.

Findings

Framework effectively models turbostratic disorder in layered materials.

Provides analytical expressions for scattering cross sections.

Applicable to various disordered layered structures.

Abstract

We present a mathematical framework to represent turbostratic disorder in materials like carbon blacks, smectites, and twisted $n$-layer graphene. In particular, the set of all possible disordered layers, including rotated, shifted, and curved layers form a stochastic sequence governed by a hidden Markov model. The probability distribution over the set of layer types is treated as an element of a Hilbert space, and using tools of Fourier analysis and functional analysis, we develop expressions for the scattering cross sections of a broad class of disordered materials.