Thermal transport in disordered nanomaterials presents unique challenges due to the absence of long-range order and the presence of structural heterogeneities. Unlike crystalline systems where phonon dynamics are well-described by periodic lattice vibrations, disordered materials such as amorphous silicon and porous nanostructures exhibit complex thermal behavior dominated by scattering, localization, and percolation effects. Understanding these mechanisms requires advanced modeling approaches that account for the interplay of disorder, defects, and nanostructuring.

Phonon localization is a critical phenomenon in disordered nanomaterials, where vibrational modes become spatially confined due to structural randomness. In amorphous materials, the lack of periodicity disrupts the propagation of extended phonon modes, leading to a dominance of diffusive and hopping-like thermal transport. Molecular dynamics simulations have shown that in amorphous silicon, the thermal conductivity is significantly reduced compared to its crystalline counterpart due to the suppression of propagating phonons. The mean free path of heat carriers in such systems is limited to a few nanometers, as evidenced by both experimental measurements and computational studies. Localized vibrational modes contribute little to thermal transport, resulting in a thermal conductivity that is nearly temperature-independent at higher temperatures, contrary to the 1/T dependence observed in crystals.

Effective medium theory (EMT) provides a useful framework for approximating the thermal properties of composite or porous nanostructures by homogenizing the heterogeneous medium. EMT treats the disordered material as a uniform effective medium with modified thermal conductivity, derived from the properties and volume fractions of its constituent phases. For porous materials, EMT models such as the Maxwell-Garnett approximation or Bruggeman’s effective medium theory are often employed. These models predict that the effective thermal conductivity decreases with increasing porosity, as the low-conductivity pores act as scattering centers. However, EMT has limitations when the characteristic length scales of heterogeneity approach or exceed the phonon mean free path, necessitating corrections for non-diffusive transport. Numerical studies on nanoporous silicon, for instance, demonstrate deviations from EMT predictions at high porosities due to increased phonon boundary scattering.

Percolation theory is another essential tool for modeling thermal transport in highly disordered or interconnected nanostructures. In materials with a mix of conducting and insulating regions, thermal percolation describes the threshold at which a continuous conductive pathway forms, enabling efficient heat transfer. Below the percolation threshold, thermal transport is dominated by isolated conductive clusters, leading to low effective conductivity. Above the threshold, the conductivity increases sharply as connected pathways emerge. For nanoporous materials, the percolation threshold depends on pore morphology and distribution. Disordered pore networks exhibit lower percolation thresholds compared to ordered arrays due to the higher likelihood of forming tortuous conductive paths. Monte Carlo simulations and finite element modeling have been used to study these effects, revealing that thermal percolation in disordered systems is highly sensitive to local connectivity and clustering.

Atomistic simulations, particularly molecular dynamics (MD), offer insights into the microscopic mechanisms of heat transfer in disordered nanomaterials. Non-equilibrium MD simulations can directly compute thermal conductivity by imposing a temperature gradient and measuring the heat flux. For amorphous silicon, such simulations reveal that short-wavelength phonons dominate thermal transport, while mid- and long-wavelength modes are heavily scattered. Green-Kubo modal analysis further decomposes the contributions of different vibrational modes, showing that diffuson-like excitations—quasi-localized vibrations with overlapping modes—play a significant role in disordered systems. These findings align with experimental observations of reduced thermal conductivity in amorphous and nanostructured materials.

Mesoscale modeling techniques, such as the Boltzmann transport equation (BTE) with spatially varying scattering rates, bridge the gap between atomistic and continuum descriptions. In disordered nanomaterials, the BTE can be modified to include frequency-dependent scattering mechanisms arising from defects, pores, and interfaces. Spectral Monte Carlo methods solve the BTE stochastically, capturing the effects of phonon dispersion and scattering anisotropy. These approaches have been applied to model thermal transport in nanoporous silicon films, where porosity-induced scattering leads to a suppression of thermal conductivity by up to two orders of magnitude compared to bulk silicon. The results agree with experimental data, validating the models’ predictive capabilities.

Machine learning (ML) has emerged as a complementary tool for predicting thermal properties of disordered nanomaterials. ML models trained on datasets generated from MD simulations or experimental measurements can rapidly estimate thermal conductivity based on structural descriptors such as pore size distribution, bond angle distortion, or density fluctuations. Neural networks have been successfully applied to predict the thermal conductivity of amorphous silicon with varying degrees of disorder, achieving good agreement with simulation results. ML approaches are particularly valuable for high-throughput screening of nanostructured materials, where traditional simulations are computationally expensive.

In summary, modeling thermal transport in disordered nanomaterials requires a multi-scale approach that integrates phonon localization, effective medium theory, and percolation concepts. Atomistic simulations provide fundamental insights into vibrational dynamics, while mesoscale models and machine learning enable efficient predictions for complex nanostructures. These methods collectively advance the understanding of heat transfer in amorphous and porous systems, guiding the design of materials for thermal insulation, thermoelectrics, and other applications where controlled thermal transport is critical. Future developments in computational techniques will further refine these models, accounting for additional complexities such as interfacial resistance and anisotropic disorder.