When we think about heat travelling through a material, we typically picture diffusive transport, a process that transfers heat from high-temperature to low-temperature as particles and molecules bump into each other, losing kinetic energy in the process. But in some materials heat can travel in a different way, flowing like water in a pipeline that – at least in principle – can be forced to move in a direction of choice. This second regime is called hydrodynamic heat transport.

Heat conduction is mediated by movement of phonons, which are collective excitations of atoms in solids, and when phonons spread in a material without losing their momentum in the process you have phonon hydrodynamics. The phenomenon has been studied theoretically and experimentally for decades, but is becoming more interesting than ever to experimentalists because it features prominently in materials like graphene, and could be exploited to guide heat flow in electronics and energy storage devices.

In a new article in Physical Review Letters, MARVEL scientists from the THEOS lab at EPFL have made a leap forward in modelling and explaining phonon hydrodynamics. Their brand new mathematical description makes the phenomenon easier to test experimentally and clarifies the physics behind it. It also points to a bizarre phenomenon that can emerge with hydrodynamic transport and by which heat can flow in reverse, from a colder region towards a hotter one.

The study’s starting point are the viscous heat equations (VHE) that were introduced in 2020 by Nicola Marzari’s group at EPFL to provide a mesoscopic description of hydrodynamic heat transport that is more suitable for simulations of devices. While the VHE enable practical numerical solutions, the physical interpretation of the components of the temperature are not immediately evident. “Our goal was to replace the numerical description with an analytical one, where hydrodynamic heat transport can be described by an actual function where you input variables and get an exact solution” says first author Enrico Di Lucente, a former member of Marzari’s EPFL lab now at Columbia University. “Having a function not only makes the problem easier to solve. It also allows you to gain more physical insight, because you see how each physical variable contributes to the result”. 

By re-expressing the VHE equations into two modified biharmonic equations (a type of partial differential equation that is often used for studying flows), the team obtained a fully analytical solution and used it to show that, in the hydrodynamic regime, the temperature emerges from two distinct contributions: one associated with the thermal compressibility of the flow and the other with its thermal vorticity. “This is an information you could not access with a numeric method” says Di Lucente. The thermal compressibility, which is formally described in this study for the first time, measures how much the phonon energy density varies in response to temperature gradients, while the thermal vorticity expresses the fluid’s spinning motion around a given point.

When applied to the in-plane section of graphite at a temperature of 70 K – that is much below standard room temperature – the equations show that a small but very surprising effect should arise. “By injecting heat at specific points, in addition to the normal heat diffusion in the center, you create vortices on the sides that push back heat from cold regions towards hot ones, a process we call thermal backflow. Thermal resistance across the device, in other words, becomes negative”.
Being able to insert such a system into consumer electronics products would have huge applications, for example hydrodynamic heat management could help prevent batteries or other devices from overheating.

“We are talking about only a couple Kelvin degrees, a very small effect” says Di Lucente. “But the equations don’t lie, the effect is there. It is up to us and to experimentalists to stabilize it enough to make it technologically appealing”. That would probably mean using a different material with a higher hydrodynamic temperature, and the very functions developed for this new study can guide towards the ideal conditions. “What we see is that the less compressible the fluid is, the more backflow you have”.

The fact that compressibility and vorticity are the fundamental variables at play also points to potential extensions of this method. “While in phonon hydrodynamics the flow is always compressible, electronic fluids are normally described as incompressible” says Di Lucente. “But there are special conditions where electron flows can be compressible too, like in plasmonics, and they are not well described by electron transport equations. Our method is a generalized description of flow that can be applied to phonons, electrons, and even magnons, that are collective magnetic excitations of particles”. 

Journal

Physical Review Letters

Funder

Swiss National Science Foundation

DOI

10.1103/g9dx-hjyn

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The quest to decipher thermal transport in organic semiconductors is both remarkable and intricate. Historically, the scientific focus has predominantly centered around charge transport, leaving researchers with a conspicuous gap in knowledge regarding how thermal energy is managed within these materials. According to Egbert Zojer, a prominent physicist leading the research, this research trajectory aimed to build a bridge between understanding both charge and thermal transport. The knowledge gleaned from this endeavor could significantly impact future innovations in organic materials, which are increasingly coveted in technological applications.

One of the most intriguing aspects of this study is the application of machine learning to decipher the intricacies of heat transport. Traditional approaches have relied heavily on empirical observations, causing researchers to miss out on potential causal connections within the materials. In this case, however, the research team opted to pursue a more fundamental path, striving for a deeper understanding of the underlying mechanics governing their thermal behaviors. By leveraging machine-learned potentials, they meticulously analyzed the distribution of heat in organic semiconductors, confronting conventional models that exclusively attributed thermal transport to the behavior of phonons.

Phonons, which represent quantized modes of vibration within a crystal lattice, have traditionally been treated as particles responsible for carrying vibrational energy. However, the research team’s findings suggest that a more intricate mechanism is at play. They uncovered evidence of tunneling transport, an additional phenomenon whereby phonons exhibit wave-like characteristics, communicating across energetic barriers within the solid matrix. This nuanced understanding of thermal transport radically reshapes the classical viewpoints surrounding the conductivity of materials, urging the scientific paradigm to embrace this more comprehensive perspective.

Equally noteworthy is the discovery that the molecular length of organic semiconductors plays a critical role in heat transport efficiency. The research team revealed that larger molecular sizes enhance tunneling effects, profoundly altering the thermal conductivity of these materials. This correlation introduces a new dimension to material design, where scientists can strategically manipulate molecular structures to optimize thermal transport tailored for specific applications. For example, in scenarios where achieving a high thermoelectric effect is crucial, focus can shift to promoting low thermal conductivity, whereas other applications may demand enhanced thermal dissipation capabilities.

Moreover, the research team’s insights extend well beyond just organic semiconductors. They propose that these findings could also be relevant to the design of metal-organic frameworks (MOFs), a class of materials recognized for their versatility and potential applications spanning from gas storage to catalysis. The intricate interplay of heat transport within MOFs makes this research invaluable, as the capability to manipulate heat conduction could significantly enhance the efficiency of various applications, paving the way for advanced innovations.

As researchers delve deeper into the mechanisms of thermal energy transport, it is clear that this study will catalyze an evolution in how materials are engineered. The traditional constraints confining our understanding of heat transfer are being dismantled, leading to new possibilities for scientists to tailor the thermal properties of materials graphically and purposefully. This transformative approach has the potential to revolutionize the fabrication of organic semiconductors and MOFs, creating opportunities for exponentially more efficient technologies in energy harvesting, electronics, and beyond.

In summary, the strides made by Egbert Zojer and his research team signify a monumental leap in the fields of materials science and thermodynamics. By integrating machine learning with fundamental research in thermodynamics, they have unraveled a newfound understanding of how heat travels in organic semiconductors which can profoundly reshape material design strategies. This research not only addresses a long-standing mystery but also illuminates a pathway toward discovering and engineering next-generation materials acclimatized to the demands of modern technology. As scientists worldwide continue to explore the implications of these findings, the future of organic semiconductors and thermal management appears more promising than ever.

This work was published in the highly regarded journal npj Computational Materials, calling attention to the importance of computational modeling in contemporary research. The merging of computational simulations with physical experiments underscores a prevailing trend: the reliance on advanced computing capabilities to tackle complex scientific challenges. As more researchers embrace these innovative methodologies, we can expect the pace of discoveries in materials science to accelerate, propelling forward our quest to harness nature’s fundamental principles for technological advancement.

Within academia and industrial circles alike, the excitement generated by these findings is palpable. The potential to utilize these principles for practical applications—be it in enhancing solar cell efficiency, creating advanced thermal insulating materials, or improving electronic devices—means that this research has implications that stretch far beyond theoretical interest, carving a niche for sustainable technology solutions that could enrich societal advancements in the years to come.

Subject of Research: Heat transport in crystalline organic semiconductors
Article Title: Heat transport in crystalline organic semiconductors: coexistence of phonon propagation and tunneling
News Publication Date: 14-Feb-2025
Web References: 10.1038/s41524-025-01514-8
References: npj Computational Materials
Image Credits: Credit: Lunghammer – TU Graz

Keywords

Organic semiconductors, thermal transport, machine learning, phonon propagation, tunneling transport, molecular length, metal-organic frameworks, material design, thermoelectric effect, computational modeling, energy efficiency