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Mathematical modelling of nanoscale heat flow and phase change

Periodic Reporting for period 1 - NanoHeat (Mathematical modelling of nanoscale heat flow and phase change)

Reporting period: 2016-09-12 to 2018-09-11

Nanotechnology is a rapidly growing interdisciplinary area, with breakthroughs having important implications in fields such as medicine, manufacturing, electronics, and energy storage. However, one of the key issues that arise when designing nanodevices, which typically have sizes that are on the order of hundreds of nanometers, is thermal management. The inability of nanoscale components to efficiently conduct heat can lead to performance degradation and ultimately device failure. Therefore, it is crucial to understand how thermal energy is transported across nanometer length scales.

Mathematical modelling of nanoscale heat transfer can provide novel insights into this process while avoiding the high costs associated with carrying out experiments. However, advances in nanotechnology have shown that traditional models of heat transfer are unable to predict the experimentally measured thermal responses of nanodevices. Thus, new mathematical models are required to describe nanoscale heat transfer.

Several experiments have shown that spherical nanoparticles melt at temperatures that are substantially lower than the usual (bulk) melting temperature of the same material. In the case of gold nanoparticles, recent reports indicate that room-temperature melting is possible. The low melting temperature of nanodevices, when combined with inefficient heat dissipation, increases the potential for catastrophic device failure. This implies there is also a need to develop models that can predict the melting temperatures of nanoscale objects and understand how nanoscale heat transport is coupled to the onset and dynamics of phase change.

The objectives of this project were to:

1. Identify extensions of Fourier's law that may be applied to nanoscale heat transfer
2. Solve extended mathematical models of heat transfer in practical scenarios
3. Through comparison with experimental data, determine the most suitable continuum model for nanoscale heat transfer
4. Develop new models of phase change on the nanoscale that (a) capture the size-dependence of material properties such as melting temperature and (b) are based on non-Fourier laws of heat transfer.
5. Solve extended mathematical models of nanoscale phase change in practical scenarios

The project members were successful at carrying out these five objectives. Continuum models of heat transfer based on the Guyer-Krumhansl (GK) equation were found to accurately predict the thermal response of nanowires. Simple expressions for the effective thermal conductivity (ETC) of nanocomponents were derived from these models. The ETC is one of the most important parameters in nanoscale heat transfer because it directly describes how well a material or component can conduct heat. New equations were derived to predict the melting temperature of nanoparticles and shown to be in excellent agreement with experimental data obtained from tin nanoparticles. New mathematical models for nanoscale phase change revealed that non-classical heat-conduction mechanisms can play a very important role in melting and should therefore be considered in future studies.
The beginning of the project focused on understanding the physics of nanoscale heat transport and the various models that exist to describe this phenomenon. After discussions with our collaborators in the physics department at the Universitat Autonoma de Barcelona, we settled on using the Guyer-Krumhansl equation to model heat transfer. Mathematical models of heat transport in cylindrical and rectangular nanowires were formulated and equations for the effective thermal conductivity (ETC) were derived. The predictions of the ETC were validated against experimental data and shown to be highly accurate. We showed that nanowires with circular cross sections are the most efficient transporters of thermal energy and should therefore be used in nanodevices that require rapid heat dissipation.

The second-half of the project focused on the study of nanoscale phase change. This involved addressing some fundamental mathematical modelling issues, such as ensuring that the governing equations predict that energy is conserved during a melting or solidification process, and determining the correct conditions to impose at a solid-liquid interface. A large part of the study of nanoscale phase change was devoted to carrying out a detailed mathematical analysis of a new theoretical model to understand how the dominant mechanisms of thermal transport change during a solidification process and influence the solidification kinetics. Other aspects of the work were more practical, and involved deriving equations that can be used to predict the melting temperature and melting behaviour of spherical nanoparticles.

The current focus is on modelling experiments which involve heating a sample using a high-frequency laser. The aim is to derive simple expressions that can be used to extract values for material properties associated with non-classical modes of heat transport. These parameter values are important when developing models to predict the thermal response of nanodevices. We are also preparing a review-like article that compares continuum models for nanoscale phase change with molecular dynamics simulations. This comparison will be used to assess the validity of each continuum model.

The work from this project led to four publications, all in Q1 journals, with four articles currently under review. Three more manuscripts are currently in preparation. Furthermore, the results from this project were presented at the 20th European Conference on Mathematics for Industry held in Budapest and featured in the blog of the European Consortium for Mathematics in Industry.
The results of this project show that relatively simple continuum models of heat transport, based on the Guyer-Krumhansl (GK) equation, are able to provide fast and accurate theoretical predictions in a number of contexts. This has many implications. Firstly, the GK equation was originally derived for solids held at low temperatures (on the order of 10 K). Our research has conclusively shown that the GK equation can be applied to room-temperature systems. As a result, this equation can form the basis for new computational tools used by engineers to carry out fast simulations of heat generation in nanodevices or devices which use nanocomponents (e.g. phones, laptops), ultimately leading to more rapid advances in nanotechnology.

Important advances were also made in understanding nanoscale phase change. We carried out the first systematic analysis of a model of phase change based on the GK equation that does not make any assumptions about the solidification kinetics. We observed that non-classical modes of heat transport that are important on the nanoscale can substantially alter the kinetics of phase change in novel ways that had not been previously reported. Furthermore, this work adds much-need clarity to the issue of the correct choice of boundary conditions in models of nanoscale phase change and showed the wrong choice can lead to physically impossible results. The need to accurately predict how nanodevices respond to their thermal environment requires physically realistic models that obey fundamental physical laws, and this project successfully established a framework for constructing such models that can be used by engineers and other researchers working in this area.
Effective thermal conductivity predictions (lines) for nanowires of diameter D
Theoretical predictions and experimental measurements of melting temperature of tin nanoparticles
Nanoparticle melting predicted with classical (Fourier) and non-classical (Maxwell-Cattaneo) laws