How Disorder Regulates Heat Flow
Designing new materials for next-generation energy technologies requires a knowledge of the relationship between atomic structure and other thermomechanical properties such as heat transport. From solar cells to nuclear reactor moderators, many technologies depend on materials whose structures lie between perfectly ordered crystals and perfectly disordered glasses. Capturing this wide spectrum of disorder in a general theory of thermomechanical properties has proven difficult.
Physicists have now devised a model of structural disorder based on the statistics of atomic arrangements, which reveals a clear relationship between disorder and the atomic vibrations that underpin thermal transport [1]. Using the relationship, the team, led by Michele Simoncelli at Columbia University, was able to explain the wide range of thermal conductivities of many structural forms of carbon. The researchers hope that this disorder model will open new pathways for assessing the macroscopic thermal conductivity of materials on the basis of their atomic structure.
In 1949, theoretical research carried out by physicist Charles Kittel suggested connections between a material’s thermal conductivity and the presence of “obstacles” that scatter atomic vibrations [2]. According to this picture, heat flows easily when obstacles are far apart, but the transport is impeded when they are close together. “This early work suggested that the distance between obstacles is determined by structural disorder: imperfections in how atoms are arranged, which can come in the form of defects, dislocations, or grain boundaries,” Simoncelli explains.
Kittel’s model was introduced in the context of quartz glass, where silicon is invariably bonded with two oxygen atoms. In carbon materials, on the other hand, a single carbon can bond with two, three, or four nearby atoms. This diversity causes carbon’s thermal conductivity to vary by 4 orders of magnitude: from extremely low values in nanoporous carbon to some of the highest known values in materials such as graphene, graphite, and diamond. Kittel’s model does not directly apply to this large spectrum of disorder, and subsequent models have not captured the fundamental physics behind structural, vibrational, and thermal properties in a unified way.
In their study, Simoncelli and colleagues revisited the problem through a new metric of structural disorder called bond-network entropy (BNE), which measures the number of different ways that atoms in a local environment are connected to each other. The size of this environment can be varied to explore the level of disorder in the atomic network.
In pure diamond, for example, every carbon atom is bonded to exactly four neighbors. This tetrahedral pattern remains identical regardless of the choice of local environment, resulting in an extremely low BNE. By contrast, in a disordered material like amorphous carbon, different local environments can exhibit significant variations in bonding patterns, angles, and distances—leading to a far higher BNE.
Crucially, the BNE is related to how atoms within a structure interact with their neighbors and thus how they can move within their environments. “This means that the BNE can be linked to atomic-vibrational patterns, which ultimately determine the conductivity,” explains Columbia team member Kamil Iwanowski.
To test their framework’s predictive power, the researchers first simulated 23 carbon polymorphs, exhibiting varying degrees of structural disorder. For each structure, they calculated the BNE value and compared it with the breadth, or “smoothness,” of its atomic-vibrational-frequency distribution. Their analysis revealed a clear inverse relationship between this smoothness and the BNE.
Using this BNE–vibration relationship, the team could establish a link to the thermal conductivity. In ordered materials, atoms usually vibrate within narrow, well-defined frequency bands, allowing heat-carrying vibrations to travel long distances without scattering. By contrast, disordered materials have blurred or poorly defined bands because of strong, disorder-mediated interactions between atomic vibrations that limit heat transport, Simoncelli says.
With the predictive power of their BNE framework, the researchers could finally explain the vast variation in the thermal conductivity of solid carbon on the basis of its degree of structural disorder. They also showed that while Kittel was broadly correct in his description of obstacles and their impact on thermal transport, his model couldn’t quantitively predict a material’s thermal conductivity from its atomic structure alone. The BNE framework solves this problem by establishing precise connections between atomic arrangements, vibrational patterns, and the length scales of thermal transport.
Owing to these insights, the team is now confident that the BNE framework could be generally applied to a wide range of solid materials beyond pure carbon. By measuring the BNE of prospective materials, engineers could predict their thermal conductivities and their suitability for extreme-temperature applications.
Even further, the BNE approach could be used to predict changes in heat-conducting properties as materials age or become damaged. For example, the atomic structure of a nuclear-reactor component can degrade over time as it is constantly bombarded by radiation—potentially impacting its thermal performance. “This would allow us to replace very expensive and challenging experiments on materials in extreme environments with fast and accurate simulations,” Simoncelli predicts.
“This work is a tour de force in its application of predictive modeling,” comments Lucas Lindsay at Oak Ridge National Laboratory in Tennessee, who has previously studied the effect of structural disorder on vibrational thermal transport. “The scope of the study is particularly striking, as it spans systems ranging from weak to strong disorder to establish clear connections between atomistic structural heterogeneity, vibrational spectra, and thermal transport.”
–Sam Jarman
Samuel Jarman is a science writer based in the UK.
References
- K. Iwanowski et al., “Bond-network entropy governs heat transport in coordination-disordered solids,” Phys. Rev. X 15, 041041 (2025).
- C. Kittel, “Interpretation of the thermal conductivity of glasses,” Phys. Rev. 75, 972 (1949).