Heat conduction in the nonlinear response regime: Scaling, boundary jumps, and negative differential thermal resistance

Scaling analysis of negative differential thermal resistance

“…It has recently been shown that the condition for the occurrence of NDTR in the φ 4 model (with an onsite potential of λx 4 i /4, where λ = 1 is the strength of the potential) can be described in terms of a scaling exponent γ > 0 that depicts the temperature dependence of the system's thermal conductivity [16]: In this nonlinear lattice, the effective thermal conductivity κ e approximately follows a scaling temperature dependence ( Fig. 1):…”

“…(3) also holds for the FPU-β model (with a β(x i+1 − x i ) 4 /4 in its potential of nearest-neighbour interaction, where β = 1 is the strength of such nonlinear interaction) with γ < 0 and t > 0 within a specific range of T . For the conventional cases of having the value of either T + or T − fixed [14][15][16][17][18][19], NDTR corresponds to…”

“…Each of these models is described by a Hamiltonian of the form where N is the total number of particles (system size) and, for the i th particle, x i is the position, p i is the instantaneous momentum, U (x i ) is the onsite potential, and V (x i+1 − x i ) is the potential of nearest-neighbour interaction. As in previous work [16], the temperature T i of the i th particle is defined as proportional to the particle's kinetic energy:…”

“…With a rich variety in their nonlinear dynamics [20,21], these models have been used as building blocks for the devices mentioned above. Recent studies [14,16] suggest that the existence of an observable regime of NDTR is conditional upon the presence of a nonlinear onsite potential as well as a small system size, the latter being attributed to boundary effects, for example thermal boundary resistance [16]. For the various lattice models mentioned above, conditions for the occurrence of NDTR have so far been studied only in terms of specific system parameters, for example in terms of the onsite potential, the system size, and the spring constant, often with the additional constraint that the temperature is kept invariant at either end of an one-dimensional lattice.…”

“…nanotube-based thermal rectifiers [10,11], thermal cloaks [12,13]), there has been much interest in understanding how NDTR can be systematically generated. In particular, studies have been carried out for various nonlinear lattices [14][15][16][17][18][19], namely the Frenkel-Kontorova (FK) model [14][15][16], the φ 4 model [16] and the Fermi-Pasta-Ulam (FPU) model [16,17]. Each of these models is described by a Hamiltonian of the form where N is the total number of particles (system size) and, for the i th particle, x i is the position, p i is the instantaneous momentum, U (x i ) is the onsite potential, and V (x i+1 − x i ) is the potential of nearest-neighbour interaction.…”