Heat conduction in the Frenkel-Kontorova model

Heat conduction is an old yet important problem. Since Fourier introduced the law bearing his name almost 200 years ago, a first-principle derivation of this simple law from statistical mechanics is still lacking. Worse still, the validity of this law in low dimensions, and the necessary and sufficient conditions for its validity are far from clear. In this paper we will review recent works on heat conduction in a simple nonintegrable model called the Frenkel-Kontorova model. The thermal conductivity of this model has been found to be finite. We will study the dependence of the thermal conductivity on the temperature and other parameters of the model such as the strength and the periodicity of the external potential. We will also discuss other related problems such as phase transitions and finite-size effects. The study of heat conduction is not only of theoretical interest but also of practical interest. We will show various recent designs of thermal rectifiers and thermal diodes by coupling nonlinear chains together. The study of heat conduction in low dimensions is also important to the understanding of the thermal properties of carbon nanotubes.     

Anomalous heat conduction and anomalous diffusion in nonlinear lattices, single walled nanotubes, and billiard gas channels

We study anomalous heat conduction and anomalous diffusion in low-dimensional systems ranging from nonlinear lattices, single walled carbon nanotubes, to billiard gas channels. We find that in all discussed systems, the anomalous heat conductivity can be connected with the anomalous diffusion, namely, if energy diffusion is sigma(2)(t)=2Dt(alpha) (01) implies an anomalous heat conduction with a divergent thermal conductivity (beta>0), and more interestingly, a subdiffusion (alpha<1) implies an anomalous heat conduction with a convergent thermal conductivity (beta<0), consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our theoretical prediction.     

Anomalous heat conduction and anomalous diffusion in one-dimensional systems

We establish a connection between anomalous heat conduction and anomalous diffusion in one-dimensional systems. It is shown that if the mean square of the displacement of the particle is =2Dt(alpha)(01) implies anomalous heat conduction with a divergent thermal conductivity (beta>0). More interestingly, subdiffusion (alpha<1) implies anomalous heat conduction with a convergent thermal conductivity (beta<0), and, consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our results.     

Thermal conductivity of hydrogenated amorphous silicon

The thermal conductivity of amorphous silicon thin films is measured in one dimension steady state condition. The experimental method is based on heating the sample front surface and monitoring the temperatures at its two sides. The experiments were carried out in conditions ensuring one-direction heat flow from top to bottom throughout the sample thickness. Sputtered a-Si:H films prepared with different conditions are used in order to investigate the dependence of thermal conductivity on material properties (i.e. hydrogen content and microstructure). The results show that, firstly, amorphous silicon is a very bad thermal conductor material. Secondly, the disorder in the film network plays an important role in thermal conduction. The highly disordered film exhibits the lowest thermal conductivity.     

应力对石墨烯热输运性质影响的分子动力学模拟

本文建立了低维薄膜材料导热模型,运用非平衡分子动力学模拟的方法,利用lanmmps软件对单层石墨烯纳米带的导热特性进行仿真分析,根据Fourier定律计算热导率,再对石墨烯纳米带的原子施加一定耦合应力场,把应力耦合作用下的石墨烯热导率与正常的石墨烯纳米带进行了对比研究,模拟数据结果表明:在石墨烯纳米带上施加耦合应力时,会导致石墨烯纳米带热导率升高,且随应力增加而增大,模拟范围内热导率升高2.61倍,并且应力方向会对热导率变化产生一定影响,这个研究为纳米尺度上石墨烯相关研究和进一步提升热导率提供了新思路.     

Temperature dependence in sputtering

We have obtained the result on the temperature dependence in sputtering of silver which is contrary to the current believes by taking steps to eliminate spurious effects due to changes of residual gas pressure and target temperature, and by determining the relevant energy range for thermal sputtering. The result i3 thought to be due basically to the thermal spike effect, and agreement is shown with a postulated thermal spike model, in which the instantaneous variation of spike temperature against thermal diffusivity is treated on the basis of Carslaw's solution of the normal heat conduction equation. The thermal diffusivity is shown to be proportionate to the thermal conductivity subject to a constant metal density in the temperature range 300–500K and a heat capacity that obeys the Dulong-Petit law which holds good for most metals at high temperatures (> θ_D). The thermal conductivity versus preheat target temperature curve matches the Makinson electronic thermal conductivity curve for metals.     

Conservation laws and associated Lie point symmetries admitted by the transient heat conduction problem for heat transfer in straight fins

Some new conservation laws for the transient heat conduction problem for heat transfer in a straight fin are constructed. The thermal conductivity is given by a power law in one case and by a linear function of temperature in the other. Conservation laws are derived using the direct method when thermal conductivity is given by the power law and the multiplier method when thermal conductivity is given as a linear function of temperature. The heat transfer coefficient is assumed to be given by the power law function of temperature. Furthermore, we determine the Lie point symmetries associated with the conserved vectors for the model with power law thermal conductivity.     

热质的运动与传递-微尺度导热中的热质动能效应

基于热质(热量的当量动质量)的概念,通过建立和分析热质的运动方程得到了反映热质动能变化的稳态导热微分方程,表明Fourier导热定律只有在热质的动能变化相对热质势能变化很小而可以忽略时才成立;在高热流密度和低温的情况下热质的动能变化不可忽略,这种动能效应表现为热流密度和温度梯度不再成线性关系.动能效应也导致Fourier导热定律不能通过热流和温度梯度准确地获得物体的导热系数,本文基于热质运动方程给出了导热系数动能效应的修正式.最后针对高热流密度和低温一维稳态导热进行了分子动力学模拟验证.     

基于热质运动概念的普适导热定律

根据爱因斯坦的质能等效关系式,热能具有的等效质量称为热质,从而在固态和气态介质中分别建立了声子气质量和热子气质量的概念.应用牛顿定律建立了含有驱动力、阻力和惯性力的热质(声子气或热子气)运动的动量守恒方程.由于热量在介质中的传递本质上就是热质(声子气和热子气)在介质中的运动,所以热质动量守恒方程就是普适的导热定律,能够统一描述各种条件下的导热规律.当热流密度不是很大从而热质惯性力可以忽略时,热质动量守恒方程就退化为傅里叶导热定律,这表明傅里叶导热定律是特殊条件下的导热定律,对于微纳尺度条件下的导热,热流密度可以极高,由速度空间变化引起的惯性力不能忽略,在稳态导热情况下也将出现非傅里叶导热,此时在计算或者实验中不能用热流密度除温度梯度求导热系数.在超快速加热条件下,必需考虑惯性力,与基于CV导热模型的波动方程相比,普适的导热定律增加了因速度空间变化引起的惯性力项,所以在介质中热波叠加时不会出现产生负温度的非物理现象,表明基于热质运动概念的普适导热定律更为合理. 关键词： 傅里叶导热定律 普适导热定律 热质运动 非傅里叶导热     

Thermal rectification in nonmetallic solid junctions: Effect of Kapitza resistance

Based on Fourier's law for heat conduction, we investigate the asymmetric heat flow in two segment rods of nonmetallic materials. Specifically, we study the effect of the Kapitza resistance at the boundary of the segments on the thermal rectification. To understand basic features of the rectification, we first develop analytical calculation for the heat currents in an ideal rod of a macroscopic length. Explicitly, this is made by assuming that the thermal conductivity of each constituent has a power-law dependence on temperature and also assuming the continuity of temperature at the boundary. Then, we introduce the temperature jump at the boundary due to the Kapitza resistance and show that this effect on the thermal rectification becomes significant as the length of the rod decreases typically to submillimeters. In particular, we find that the temperature jump yields a finite rectification even when no asymmetry is predicted in the heat currents from the continuity of temperature at the junction. The obtained results have an important implication for the analysis of the thermal rectification of a rod consisting of semiconductors Ge and Si.     

Breakdown of Fourier's law in nanotube thermal conductors

We present experimental evidence that the room temperature thermal conductivity (kappa) of individual multiwalled carbon and boron-nitride nanotubes does not obey Fourier's empirical law of thermal conduction. Because of isotopic disorder, kappa's of carbon nanotubes and boron-nitride nanotubes show different length dependence behavior. Moreover, for these systems we find that Fourier's law is violated even when the phonon mean free path is much shorter than the sample length.     

Computer Simulation Study of Thermal Conduction in 1D Chains of Anharmonic Oscillators

In this work thermal conduction in one-dimensional(1D) chains of anharmonic oscillators are studied using computer simulation.The temperature profile,heat flux and thermal conductivity are investigated for chain length N = 100,200,400,800 and 1600.In the computer simulation anharmonicity is introduced due to Fermi-Pasta-Ulam-β(FPU-β) model.For substrate interaction,an onsite potential due to Frenkel-Kontorova(FK) model has been used.Numerical simulations demonstrate that temperature gradient scales behave as N-1 linearly with the relation J = 0.1765/N.For the thermal conductivity K,KN to N obey the linear relation of the type KN = 0.8805N.It is shown that thermal transport is dependent on phonon-phonon interaction as well as phonon-lattice interaction.The thermal conductivity increases linearly with increase inanharmonicity and predicts relation κ = 0.133 + 0.804β.It is also concluded that for higher value of the strength of the onsite potential system tends to a thermal insulator.     

导热优化:热耗散与最优导热系数场

本文讨论了通过重新布置场内的导热系数分布来强化热传导的导热优化问题。针对边界上传热量一定的任意几何区域的稳态导热,在全场的导热系数积分为定值的条件下,以最小热耗散作为目标,利用变分方法导出了导热优化的基本准则;导热系数与局部热流密度成正比。该准则指导下的导热优化过程可获得热耗散最小的导热系数分布。实例证明了该方法是有效且合理的。     

含有格点势的一维Fibonacci链热传导性质的研究

用传输矩阵的方法,研究了格点势(on-site势)对一维Fibonacci链的热传导性质(透射系数、Lyapunov指数及热导率κ)的影响.研究结果表明:当固定原子质量比和力作用常数比时,随着格点势的增大,低频区域的透射系数减小,对应的Lyapunov指数增大,透射谱向高频方向移动.同时,格点势越大,同样大小的体系对应的κ越小,当格点势足够大时,κ会趋近零.在热导率κ对振动频率ω²作图中,κ呈现台阶式缓慢上升的趋势,且在高频     

Two-channel Kondo effect in glasslike ThAsSe

We present low-temperature heat and charge transport as well as caloric properties of a ThAsSe single crystal. An extra -AT(1/2) term in the electrical resistivity, independent of magnetic fields as high as 14 T, provides evidence for an unusual scattering of conduction electrons. Additionally, both the thermal conductivity and the specific heat show a glass-type temperature dependence which signifies the presence of tunneling states. These observations apparently point to an experimental realization of a two-channel Kondo effect derived from structural two-level systems.     

Normal heat conduction in a chain with a weak interparticle anharmonic potential

We analytically study heat conduction in a chain with an interparticle interaction V(x)= lambda[1-cos(x)] and harmonic on-site potential. We start with each site of the system connected to a Langevin heat bath, and investigate the case of small coupling for the interior sites in order to understand the behavior of the system with thermal reservoirs at the boundaries only. We study, in a perturbative analysis, the heat current in the steady state of the one-dimensional system with a weak interparticle potential. We obtain an expression for the thermal conductivity, compare the low and high temperature regimes, and show that, as we turn off the couplings with the interior heat baths, there is a "phase transition": Fourier's law holds only at high temperatures.     

Heat Conduction in the α−β Fermi–Pasta–Ulam Chain

Recent simulation results on heat conduction in a one-dimensional chain with an asymmetric inter-particle interaction potential and no onsite potential found non-anomalous heat transport in accordance to Fourier’s law. This is a surprising result since it was long believed that heat conduction in one-dimensional systems is in general anomalous in the sense that the thermal conductivity diverges as the system size goes to infinity. In this paper we report on detailed numerical simulations of this problem to investigate the possibility of a finite temperature phase transition in this system. Our results indicate that the unexpected results for asymmetric potentials is a result of insufficient chain length, and does not represent the asymptotic behavior.     

Influence of heat bath on the heat conductivity in disordered anharmonic chain

We study heat conduction in a one-dimensional disordered anharmonic chain with arbitrary heat bath by using extended Ford, Kac and Mazur (FKM) formulation, which satisfy the fluctuation-dissipation theorem. A simple formal expression for the heat conductivity κ is obtained, from which the asymptotic system-size (N) dependence is extracted. It shows κ∼N^α. As a special case we give the expression that κ∼N^1/2 for free boundaries, and κ∼ N^-1/2 for fixed boundaries, from which we can get the conclusion that the momentum conservation is a key factor of the anomalous heat conduction. Comparing with different ∇T, the heat conductivity shows large difference between the linear system and the nonlinear system.     

Anomalous heat conduction and anomalous diffusion in low dimensional nanoscale systems

Heat conduction is an important energy transport process in nature. Phonon is the major energy carrier for heat in semiconductors and dielectric materials. In analogy to Ohm’s law of electrical conduction, Fourier’s law is the fundamental law of heat conduction in solids. Although Fourier’s law has received great success in describing macroscopic heat conduction in the past two hundred years, its validity in low dimensional systems is still an open question. Here we give a brief review of the recent developments in experimental, theoretical and numerical studies of heat conduction in low dimensional systems, including lattice models and low dimensional nanostructures such as nanowires, nanotubes and graphene. We will demonstrate that phonons transport in low dimensional systems superdiffusively, which leads to a size dependent thermal conductivity. In other words, Fourier’s law is not applicable in low dimensional structures.     

Thermal conductivity for single-walled carbon nanotubes from Einstein relation in molecular dynamics

Equilibrium molecular dynamics based Einstein relation with an appropriate definition for integrated heat current (i.e., with modified energy moment) are combined to quantify the thermal conductivity of individual single-walled carbon nanotubes, armchair, zigzag and chiral tubes. The thermal conductivity has been investigated as a function of three parameters, tube radius, length and chirality at and near room temperature with Brenner potential model. Thermal conductivity is found to have unusually high value and varies with radius, length and chirality of tubes. Also the thermal conductivity at temperature range from 50 to 100 K is found to have a maximum value. For 12.1 nm tube length, the thermal conductivity has converging trend which its value dependents on the tube radius and chirality. Tubes with large radius have lower values of thermal conductivity. Furthermore, the results show that armchair tubes have large values of the thermal conductivity comparing with zigzag and chiral tubes. It seems possible to uncover carbon nanotubes thermal properties based on measurements having heat dependence by adding another methods for calculations.