Heat conduction in nanoscale materials: a statistical-mechanics derivation of the local heat flux

Size effects in energy transport between thermal contacts mediated by nanoparticles

We investigate size effects in phononic energy transport in a system of two nanoparticles interconnected by a molecule and attached to thermal contacts also by molecules. In the considered closed system, the nanoparticles and contacts are described by ensembles of finite numbers of harmonic oscillators within the Drude-Ullersma model. The macroscopic character of the contacts is simulated by a large value of the ratio Δ/Δ_{B}=n (n>100) of mode spacings Δ and Δ_{B} corresponding to the nanoparticles and contacts, respectively. Quasistatic energy transport on the timescale Δ^{-1} is investigated. Equations describing the dynamics of the averaged eigenmode energies that belong to the nanoparticles and contacts are derived and solved. The resulting expressions for the energy current exiting (entering) the contacts as well as the energy current between the nanoparticles are obtained and investigated. The latter current accounts for energy accumulation by (depletion from) the nanoparticles. The finite size effects result in reversibility features and peculiarities at time moments t=2πℓΔ^{-1} for non-negative integers ℓ. They are qualitatively the same as in a previously studied system of two equal nanoparticles mediated by a molecule, despite the presence of the macroscopic contacts. The thermal conductance of the whole nanojunction is derived and explored. The energy currents and thermal conductance of the nanojunction in a case when its parameters are known from the experiment are computed using the developed model.

The entropy of a complex molecule

Entropy is a central concept in the theory of coarse-graining. Through Einstein's formula, it provides the equilibrium probability distribution of the coarse-grained variables used to describe the system of interest. We study with molecular dynamics simulations the equilibrium probability distribution of thermal blobs representing at a coarse-grained level star polymer molecules in melt. Thermal blobs are characterized by the positions and momenta of the centers of mass, and internal energies of the molecules. We show that the entropy of the level of description of thermal blobs can be very well approximated as the sum of the thermodynamic entropy of each single molecule considered as isolated thermodynamic systems. The entropy of a single molecule depends on the intrinsic energy, involving only contributions from the atoms that make the molecule and not from the interactions with atoms of other molecules.

Local momentum and heat fluxes in transient transport processes and inhomogeneous systems

This work examines existing formalisms for the derivation of microscopic momentum and heat fluxes. Both analytical and simulation results are provided to show that the widely used flux formulas are not applicable to transient transport processes or highly inhomogeneous systems, e.g., materials with atomically sharp interfaces. A method is formulated for formally deriving microscopic momentum and heat fluxes through the integral representation of conservation laws. The resulting flux formulas are mathematically rigorous, fully consistent with the physical concepts of momentum and heat fluxes, and applicable to nonequilibrium transient processes in atomically inhomogeneous systems with general many-body forces.

Modeling the quasistatic energy transport between nanoparticles

We consider phononic energy transport between nanoparticles mediated by a quantum particle. The nanoparticles are considered as thermal reservoirs described by ensembles of finite numbers of harmonic oscillators within the Drude-Ullersma model having, in general, unequal mode spacings Δ(1) and Δ(2), which amount to different numbers of atoms in the nanoparticles. The quasistatic energy transport between the nanoparticles on the time scale t∼1/Δ(1,2) is investigated using the generalized quantum Langevin equation. We find that double degeneracy of system's eigenfrequencies, which occurs in the case of identical nanoparticles, is removed when the mode spacings become unequal. The equations describing the dynamics of the averaged eigenmode energies are derived and solved, and the resulting expression for the energy current between the nanoparticles is obtained and explored. Unlike the case when the thermodynamic limit is assumed resulting in time-independent energy current, finite-size effects result in temporal behavior of the energy current that evinces reversibility features combined with decay and possesses peculiarities at time moments t=2πn/Δ(1)+2πm/Δ(2) for non-negative integers n and m. When Δ(1,2)→0, an expression for the heat current obtained previously under assumption of the thermodynamic limit is reproduced. The energy current between two platinum nanoparticles mediated by a carbon oxide molecule is considered as an application of the developed model.