Temperature dependence of thermal conductivity in 1D nonlinear lattices

We examine the temperature dependence of thermal conductivity of one dimensional nonlinear (anharmonic) lattices with and without on-site potential. It is found from computer simulation that the heat conductivity depends on temperature via the strength of nonlinearity. Based on this correlation, we make a conjecture in the effective phonon theory that the meanfree-path of the effective phonon is inversely proportional to the strength of nonlinearity. We demonstrate analytically and numerically that the temperature behavior of the heat conductivity κ ∝ 1/T is not universal for 1D harmonic lattices with a small nonlinear perturbation. The computer simulations of temperature dependence of heat conductivity in general 1D nonlinear lattices are in good agreements with our theoretic predictions. Possible experimental test is discussed. The role of nonlinearity in the dynamics of one dimensional (1D) nonlinear (anharmonic) lattices has attracted attention for many decades, for instance, the ergodicity problem in the Fermi-Pasta-Ulam (FPU) chains inducted by nonlinear interactions [1,2], solitons in nonlinear partial differential equations [3] and discrete breathers in nonlinear lattices [4, 5], to name just a few. Recent years have witnessed increasing studies on the role of nonlinearity in heat conduction in low dimensional systems (See Ref. [6] and the references therein). The fundamental question is whether the non-integrability or chaos is an essential or a sufficient condition for the heat conduction to obey the Fourier law. From computer simulations, it is found that in some 1D nonlinear lattices such as the FrenkelKontorova (FK) model and the φ model, the heat conductivity is size independent [7–9]. Thus the heat conduction in these models obeys the Fourier’s law. This transport behavior is called normal heat conduction. Whereas in some other nonlinear lattices such as the FPU and alike models, the heat conduction exhibits anomalous behavior [10], namely the heat conductivity κ diverges with the system size N as κ ∼ N . Studies in past years have focused on the physical origin and the value of the divergent exponent δ [10–14]. It is found that the anomalous heat conduction is due to the anomalous diffusion and a quantitative connection between them has been established [15]. Most recently, a very general effective phonon theory has been proposed to describe the normal and anomalous heat conduction under the same framework [16]. As a by-product of the study of heat conduction in low dimensional systems, nonlinearity (anharmonicity) has been found very useful in controlling heat flow. Because of the nonlinearity, the lattice vibrational spectrum depends on temperature. This property has been used to design the thermal rectifiers/diodes [17–22] and thermal transistors [23]. Inspired by the two segment theoretical models [18, 19], Chang et al [24] has built the first solid state thermal rectifier with a single walled carbon nanotube and boron nitride nanotube, which indicates the opening of a new research field controlling heat flow at microscopic level by using nonlinearity. However, in contrast to the size-dependence, the temperature dependence of heat conductivity and its relationship with the nonlinearity have not yet been studied systematically even though this problem is very fundamental and very important from the experimental point of view. In fact, for low dimensional nanoscale systems such as nanotube and nanowires etc, to measure the temperature dependence of thermal conductivity for a fixed length sample is much easier than that for the size dependence of ther-