A new unified theory for heat transport accurately describes a wide range of materials – from crystals and polycrystalline solids to alloys and glasses – and allows them to be treated in the same way for the first time. The methodology, which is based on the Green-Kubo theory of linear response and concepts from lattice dynamics, naturally accounts for quantum mechanical effects and thus allows for the predictive modelling of heat transport in glasses at low temperature – a feat never achieved before, say the researchers who developed it. It will be important for better understanding and designing heat transporting devices in a host of applications, from heat management in high-power electronics, batteries and photovoltaics to thermoelectric energy harvesting and solid-state cooling. It might even help describe heat flow in planetary systems.
“Heat transport is the fundamental mechanism through which thermal equilibrium is reached,” explains Stefano Baroni of the Scuola Internazionale Superiore di Studi Avanzati (SISSA) in Trieste, Italy, who led this research effort. “It can also be thought of as the most fundamental manifestation of irreversibility in nature – as heat flows from warm areas in the same system to cooler ones as time flows from the past to the future (the ‘arrow of time’). What is more, many modern technologies rely on our ability to control heat transport.”
However, despite its importance, heat transport is still poorly understood and it is difficult to simulate the heat transport of materials because of this lack of understanding. To overcome this knowledge gap, researchers employ various simulation techniques based on diverse physical assumptions and approximations for different classes of material – crystals on one hand and disordered solids and liquids on the other.
“No all-encompassing theory existed before our work and researchers did not know how to properly simulate a system that is comprised of different classes of materials (a crystal or glass, for instance) or a system that is not easily classified (a defective or partially disordered crystal, for example),” says Baroni. “And in the case of glasses at low temperature, in which quantum effects and disorder co-exist, no simulation method was available at all.”
Filling the gaps
Our new approach fills these gaps in a theoretically rigorous and practically viable way, he tells Physics World.
Until now, there were essentially two independent (and seemingly unrelated) approaches to describing heat transport, he explains. The first is the Boltzmann-Peierls kinetic approach and the second the Green and Kubo linear-response approach. The former assumes that heat is carried by quasi-particles known as phonons (which are the quanta that carry sound waves). These particles have a well-defined energy and velocity and in this approach the heat conductivity is proportional to the product of the square of the phonon velocity and its mean free path (that is, the distance it travels before it bounces off an impurity or another phonon).
“For this approach to apply, however, the mean free path needs to be much larger than the average distance between neighbouring atoms so that we can meaningfully define and compute the phonon velocity,” explains Baroni. “This is the case for crystalline solids, but not for glasses and disordered solids in general, or indeed for liquids, for which the very concept of quasi-particle beaks down.”
The Green-Kubo method, for its part, is, in principle, applicable to a wider range of materials, but in reality, it is extremely inefficient at low and intermediate temperatures (generally speaking, below half the melting temperature of a material). And while, in theory, it can account for quantum effects, no practical techniques are available to numerically simulate these effects.
“Our new approach unifies these hitherto distinct approaches into the same methodology that is applicable to all solids – be they crystalline or disordered/glassy,” says Baroni. “It also actually works better at temperatures in the regime where current implementations of the Green-Kubo theory fail.”
The Green-Kubo theory of linear response
At equilibrium, a system carries no heat current on average, but small current “flickers” do occur because of thermal fluctuations. “The Green-Kubo theory of linear response states that the heat conductivity is proportional to the product of the square of the (average) magnitude of the current fluctuations multiplied by the (average) time it takes them to fade off,” explains Baroni. “The fluctuations have to fade off otherwise the heat energy would be persistent and not flickering.”
Usually, the Green-Kubo theory requires that the equations of motion for a model system comprising several hundred to a few thousand atoms are solved (using molecular dynamics) for a time ranging from a few hundred picoseconds to a few nanoseconds, he says. The longer it takes for a system to reach equilibrium, the longer the simulation time must be. And the lower the temperature, the more ordered the system is and the longer it takes for the system to come to equilibrium. This is why Green-Kubo molecular dynamics is not the method of choice for ordered crystals at low temperature and the Boltzmann-Peierls approach is favoured instead – even though this approach does not apply to disordered systems.
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The harmonic approximation
“Our methodology avoids these difficulties by analytically solving the equations of motion (either classical or quantum mechanical) in the so-called harmonic approximation, which is the basis of the theory of lattice dynamics, and means that we can now thus simulate quantum effects in thermal transport without making use of molecular dynamics,” explains Baroni. “The harmonic approximation assumes that the forces acting on individual atoms are proportional to their displacements from the positions of mechanical equilibrium that would be appropriate at zero temperature.
“As the magnitude of these displacements decreases when the temperature decreases, this approximation becomes increasingly more accurate as the temperature drops. We have shown that the Green-Kubo formula for the thermal conductivity can be analytically computed within this approximation and that it yields the same results as the Boltzmann-Peierls method in the case of a perfect crystal. But, it also happily provides well-defined results for glasses (for which the Boltzmann-Peierls approach would not work).”
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Disordered and crystalline solids now on the same footing
Our scheme allows us to deal with disordered and crystalline solids on the same footing for the first time and with the same level of accuracy, he states. “This will allow scientists and engineers to understand and design heat transport for a wide range of technological applications that were thought unfeasible thus far. Such applications include: thermoelectric energy harvesting; solid-state cooling; thermal insulation; and thermal barrier coatings (all of which require extremely low thermal conductivity); and heat management in high-power electronics; batteries; and photovoltaics (all of which require high thermal conductivity).
“The materials employed in these applications are nanostructured, polycrystalline, highly defective or even glassy. We can now study them all with high accuracy, within a unified and practicable framework.”
The researchers reporting their work in Nature Communications 10.1038/s41467-019-11572-4, say they will now be looking for a methodology that allows them to treat quantum effects beyond the harmonic approximation – that is, at higher temperatures.
