Rapid progress in the synthesis and processing of materials with structure on nanometer length scales has created a demand for greater scientific understanding of thermal transport in nanoscale devices, individual nanostructures, and nanostructured materials. This review emphasizes developments in experiment, theory, and computation that have occurred in the past ten years and summarizes the present status of the field. Interfaces between materials become increasingly important on small length scales. The thermal conductance of many solid–solid interfaces have been studied experimentally but the range of observed interface properties is much smaller than predicted by simple theory. Classical molecular dynamics simulations are emerging as a powerful tool for calculations of thermal conductance and phonon scattering, and may provide for a lively interplay of experiment and theory in the near term. Fundamental issues remain concerning the correct definitions of temperature in nonequilibrium nanoscale systems. Modern Si microelectronics are now firmly in the nanoscale regime—experiments have demonstrated that the close proximity of interfaces and the extremely small volume of heat dissipation strongly modifies thermal transport, thereby aggravating problems of thermal management. Microelectronic devices are too large to yield to atomic-level simulation in the foreseeable future and, therefore, calculations of thermal transport must rely on solutions of the Boltzmann transport equation; microscopic phonon scattering rates needed for predictive models are, even for Si, poorly known. Low-dimensional nanostructures, such as carbon nanotubes, are predicted to have novel transport properties; the first quantitative experiments of the thermal conductivity of nanotubes have recently been achieved using microfabricated measurement systems. Nanoscale porosity decreases the permittivity of amorphous dielectrics but porosity also strongly decreases the thermal conductivity. The promise of improved thermoelectric materials and problems of thermal management of optoelectronic devices have stimulated extensive studies of semiconductor superlattices; agreement between experiment and theory is generally poor. Advances in measurement methods, e.g., the 3ω method, time-domain thermoreflectance, sources of coherent phonons, microfabricated test structures, and the scanning thermal microscope, are enabling new capabilities for nanoscale thermal metrology.

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Nanoscale thermal transport

Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.

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The Kuramoto model: A simple paradigm for synchronization phenomena

Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

http://dsec.pku.edu.cn/~tieli/notes/numer_anal/MCQMC_Caflisch.pdf

Monte Carlo and quasi-Monte Carlo methods

This report documents a Fortran computer code package that is used for the evaluation of gas-phase multicomponent viscosities, thermal conductivities, diffusion coefficients, and thermal diffusion coefficients. The package is in two parts. The first is a preprocessor that computes polynomial fits to the temperature dependent parts of the pure species viscosities and binary diffusion coefficients. The coefficients of these fits are passed to a library of subroutines via a linking file. Then, any subroutine from this library may be called to return either pure species properties or multicomponent gas mixture properties. This package uses the chemical kinetics package CHEMKIN, and transport property subroutines are designed to be used in conjunction with the CHEMKIN subroutine library. This package supersedes a previously-written transport property code package in which we used certain mixture averaging rules to compute mixture properties rather than the full multicomponent formulation.

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A fortran computer code package for the evaluation of gas-phase, multicomponent transport properties

This paper reviews applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions. We first summarize the available simulation methods for colloidal suspensions together with some of the important applications of these methods, and then describe results from lattice-gas and lattice-Boltzmann simulations in more detail. The remainder of the paper is an update of previously published work,(69, 70) taking into account recent research by ourselves and other groups. We describe a lattice-Boltzmann model that can take proper account of density fluctuations in the fluid, which may be important in describing the short-time dynamics of colloidal particles. We then derive macro-dynamical equations for a collision operator with separate shear and bulk viscosities, via the usual multi-time-scale expansion. A careful examination of the second-order equations shows that inclusion of an external force, such as a pressure gradient, requires terms that depend on the eigenvalues of the collision operator. Alternatively, the momentum density must be redefined to include a contribution from the external force. Next, we summarize recent innovations and give a few numerical examples to illustrate critical issues. Finally, we derive the equations for a lattice-Boltzmann model that includes transverse and longitudinal fluctuations in momentum. The model leads to a discrete version of the Green–Kubo relations for the shear and bulk viscosity, which agree with the viscosities obtained from the macro-dynamical analysis. We believe that inclusion of longitudinal fluctuations will improve the equipartition of energy in lattice-Boltzmann simulations of colloidal suspensions.

Lattice-Boltzmann Simulations of Particle-Fluid Suspensions

SINCE the days when the kinetic theory of gases was founded by Clausius, Maxwell and Boltzmann, great changes have taken place in the scope and bearing of the descendent theory. Taking a wide view of its present field, the qualification “of gases” has become a misnomer. One can see that the ultimate scope of the descendent of the kinetic theory of gases must be the whole field of the properties of matter in bulk, derived from the atomic constitution of matter and from the properties of atoms and their interactions. The present scope of the theory falls short of such inclusiveness, but not so far short as to leave any doubts for the future. The Mathematical Theory of Non-Uniform Gases An Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases. By Prof. Sydney Chapman and Dr. T. G. Cowling. Pp. xxiii + 404. (Cambridge: At the University Press, 1939.) 30s. net.

R. H. Fowler

Papers

The Mathematical Theory of Non-Uniform Gases