Heat conduction in one-dimensional lattices with on-site potential

Approaching the perfect diode limit through a nonlinear interface

We consider a system formed by two different segments of particles, coupled to thermal baths, one at each end, modeled by Langevin thermostats. The particles in each segment interact harmonically and are subject to an on-site potential for which three different types are considered, namely, harmonic, ϕ^{4}, and Frenkel-Kontorova. The two segments are nonlinearly coupled, between interfacial particles, by means of a power-law potential with exponent μ, which we vary, scanning from subharmonic to superharmonic potentials, up to the infinite-square-well limit (μ→∞). Thermal rectification is investigated by integrating the equations of motion and computing the heat fluxes. As a measure of rectification, we use the difference of the currents, resulting from the interchange of the baths, divided by their average (all quantities taken in absolute value). We find that rectification can be optimized by a given value of μ that depends on the bath temperatures and details of the chains. But, regardless of the type of on-site potential considered, the interfacial potential that produces maximal rectification approaches the infinite square well (μ→∞) when reducing the average temperature of the baths. Our analysis of thermal rectification focuses on this regime, for which we complement numerical results with heuristic considerations.

Kapitza resistance at a domain boundary in linear and nonlinear chains

We explore Kapitza thermal resistance on the boundary between two homogeneous chain fragments with different characteristics. For a linear model, an exact expression for the resistance is derived. In the generic case of frequency mismatch between the domains, the Kapitza resistance is well defined in the thermodynamic limit. At the same time, in the linear chain, the resistance depends on the thermostat properties and therefore is not a local property of the considered domain boundary. Moreover, if the temperature difference at the ends of the chain is fixed, then neither the temperature drop at the domain boundary nor the heat flux depend on the system size; for the normal transport, one expects the scaling N^{-1} for both. For specific assessment, we consider the case of an isotopic boundary-only the masses in different domains are different. If the domains are nonlinear, but integrable (Toda lattice, elastically colliding particles), the anomalies are similar to the case of linear chain, with the addition of well-articulated thermal dependence of the resistance. For the case of elastically colliding particles, this dependence follows a simple scaling law R_{k}∼T^{-1/2}. For Fermi-Pasta-Ulam domains, both the temperature drop and the heat flux decrease with the chain length, but with different exponents, so the resistance vanishes in the thermodynamic limit. For the domains comprised of rotators, the thermal resistance exhibits the expected normal behavior.

Geometry-induced local thermal current from cold to hot in a classical harmonic system

The second law of thermodynamics requires the overall thermal current to flow from hot to cold. However, it does not forbid a local thermal current from flowing from cold to hot. By coupling a harmonic system of three masses connected by a few springs to two Langevin reservoirs at different temperatures, a local atypical thermal current is found to flow from cold to hot in the steady state while the overall thermal current is still from hot to cold. The direction of the local thermal current can be tuned by the mass, spring constant, and system-reservoir coupling. The local thermal current can vanish if the parameters are tuned to proper values. We also consider nonlinear effect from the system-substrate coupling and find that the local atypical thermal current survives in the presence of the nonlinear potential. Moreover, the local atypical thermal current is robust against asymmetry of the system-reservoir coupling, inhomogeneity of the nonlinear potential, and additions of more masses and springs. In molecular or nanomechanical systems where the setup may find its realization, the direction of the local thermal current may be controlled by mechanical or electromagnetic means, which may lead to applications in information storage.

Scattering of lattice solitons and decay of heat-current correlation in the Fermi-Pasta-Ulam-α-β model

As is well known, solitons can be excited in nonlinear lattice systems; however, whether, and if so, how, this kind of nonlinear excitation can affect the energy transport behavior is not yet fully understood. Here we study both the scattering dynamics of solitons and heat transport properties in the Fermi-Pasta-Ulam-α-β model with an asymmetric interparticle interaction. By varying the asymmetry degree of the interaction (characterized by α), we find that (i) for each α there exists a momentum threshold for exciting solitons from which one may infer an α-dependent feature of probability of presentation of solitons at a finite-temperature equilibrium state and (ii) the scattering rate of solitons is sensitively dependent on α. Based on these findings, we conjecture that the scattering between solitons will cause the nonmonotonic α-dependent feature of heat conduction. Fortunately, such a conjecture is indeed verified by our detailed examination of the time decay behavior of the heat current correlation function, but it is only valid for an early time stage. Thus, this result may suggest that solitons can have only a relatively short survival time when exposed in a thermal environment, eventually affecting the heat transport in a short time.

Long-time instability in the Runge-Kutta algorithm for a Nosé-Hoover heat bath of a harmonic chain and its stabilization

In this paper, we investigate the Runge-Kutta algorithm for the Nosé-Hoover heat bath of a harmonic chain. The Runge-Kutta algorithm is found to be unstable in long-time calculations, with the system temperature growing exponentially. The growth rate increases if time step size is chosen larger. By analyzing the Fourier spectra in both space (wave number) and time (frequency), we discover that the growth is caused by spurious energy accumulation, particularly at the largest wave number. Such accumulation may be explained by von Neumann analysis for an infinite chain, with the nonlinear heat bath being ignored. Furthermore, we propose to add a filter to remove excessive energy, which effectively stabilizes the algorithm.

Stretch diffusion and heat conduction in one-dimensional nonlinear lattices

For heat conduction in one-dimensional (1D) nonlinear Hamiltonian lattices, it has been known that conserved quantities play an important role in determining the actual heat conduction behavior. In closed or microcanonical Hamiltonian systems, the total energy and stretch are always conserved. Depending on the existence of external on-site potential, the total momentum can be conserved or not. All the momentum-conserving lattices have anomalous heat conduction except the 1D coupled rotator lattice. It was recently claimed that "whenever stretch (momentum) is not conserved in a 1D model, the momentum (stretch) and energy fields exhibit normal diffusion." The stretch in a coupled rotator lattice was also argued to be nonconserved due to the requirement of a finite partition function, which enables the coupled rotator lattice to fulfill this claim. In this work, we will systematically investigate stretch diffusion and heat conduction in terms of energy diffusion for typical 1D nonlinear lattices. Contrary to what was claimed, no clear connection between conserved quantities and heat conduction can be established. The actual situation might be more complicated than what was proposed.

Dynamical thermalization of Frenkel-Kontorova model in the thermodynamic limit

We study numerically the process of dynamical thermalization in the Frenkel-Kontorova (FK) model with weak nonlinearity. The total energy has initially equidistributed among some of the lowest frequency linear modes. It is found that the energy transfers continuously to the high-frequency modes and finally evolves towards energy equipartition in the FK model. However, the metastable state, which was found in Fermi-Pasta-Ulam (FPU) model and φ(4) model in a relatively short time scale, is not found in the FK model. We further perform a very accurate systematic study of the equipartition time T(eq) as functions of the particle number N, the nonlinear parameter β, and the energy density ɛ. In the thermodynamic limit, the dependence of T(eq) on β and ɛ is found to display a power law behavior: T(eq)∝β(a)ɛ(b). The exponents a and b are numerically found to be approximately -2.0 and 1.43. This scaling law is also quite different from those of the FPU-β model and φ(4) model.

Temperature-dependent thermal conductivities of one-dimensional nonlinear Klein-Gordon lattices with a soft on-site potential

The temperature-dependent thermal conductivities of one-dimensional nonlinear Klein-Gordon lattices with soft on-site potential (soft-KG) are investigated systematically. Similarly to the previously studied hard-KG lattices, the existence of renormalized phonons is also confirmed in soft-KG lattices. In particular, the temperature dependence of the renormalized phonon frequency predicted by a classical field theory is verified by detailed numerical simulations. However, the thermal conductivities of soft-KG lattices exhibit the opposite trend in temperature dependence in comparison with those of hard-KG lattices. The interesting thing is that the temperature-dependent thermal conductivities of both soft- and hard-KG lattices can be interpreted in the same framework of effective phonon theory. According to the effective phonon theory, the exponents of the power-law dependence of the thermal conductivities as a function of temperature are only determined by the exponents of the soft or hard on-site potentials. These theoretical predictions are consistently verified very well by extensive numerical simulations.

Thermal conductivity of molecular chains with asymmetric potentials of pair interactions

We provide molecular-dynamics simulation of heat transport in one-dimensional molecular chains with different interparticle pair potentials. We show that the thermal conductivity is finite in the thermodynamic limit in chains with the potentials that allow for bond dissociation. The Lennard-Jones, Morse, and Coulomb potentials are such potentials. The convergence of the thermal conductivity is provided by phonon scattering on the locally strongly stretched loose interatomic bonds at low temperature and by the many-particle scattering at high temperature. On the other hand, chains with a confining pair potential, which does not allow for bond dissociation, possess anomalous thermal conductivity, diverging with the chain length. We emphasize that chains with a symmetric or asymmetric Fermi-Pasta-Ulam potential or with combined potentials, containing a parabolic and/or a quartic confining potential, all exhibit anomalous heat transport.

Mechanical control of heat conductivity in molecular chains

We discuss a possibility to control heat conductivity in molecular chains by means of external mechanical loads. To illustrate such possibilities we consider first well-studied one-dimensional chain with degenerate double-well potential of the nearest-neighbor interaction. We consider varying lengths of the chain with fixed number of particles. Number of possible energetically degenerate ground states strongly depends on the overall length of the chain, or, in other terms, on average length of the link between neighboring particles. These degenerate states correspond to mechanical equilibria; therefore, one can say that formation of such structures mimics a process of plastic deformation. We demonstrate that such modification of the chain length can lead to quite profound (almost fivefold) reduction of the heat conduction coefficient. Even more profound effect is revealed for a model with a single-well nonconvex potential. It is demonstrated that in a certain range of constant external forcing, this model becomes effectively double-well and has a multitude of possible states of equilibrium for fixed value of the external load. Due to this degeneracy, the heat-conduction coefficient can be reduced by two orders of magnitude. We suggest a mechanical model of a chain with periodic double-well potential, which allows control of the heat transport. The models considered may be useful for description of heat transfer in biological macromolecules and for control of the heat transport in microsystems. The possibility of the heat transport control in more realistic three-dimensional systems is illustrated by simulation of a three-dimensional model of polymer α-helix. In this model, the mechanical stretching also brings about the structural inhomogeneity and, in turn, to essential reduction of the heat conductivity.

Crossover from Fermi-Pasta-Ulam to normal diffusive behavior in heat conduction through open anharmonic lattices

We study heat conduction in one-, two-, and three-dimensional anharmonic lattices connected to stochastic Langevin heat baths. The interatomic potential of the lattices is double-well type, i.e., V(DW)(x)=k(2)x(2)/2+k(4)x(4)/4 with k(2)<0 and k(4)>0. We observe two different temperature regimes of transport: a high-temperature regime where asymptotic length dependence of nonequilibrium steady state heat current is similar to the well-known Fermi-Pasta-Ulam lattices with an interatomic potential, V(FPU)(x)=k(2)x(2)/2+k(4)x(4)/4 with k(2),k(4)>0, and a low-temperature regime where heat conduction is most likely diffusive normal, satisfying Fourier's law. We present our simulation results for different temperature regimes in all dimensions.

Nonstationary heat conduction in one-dimensional models with substrate potential

The paper investigates nonstationary heat conduction in one-dimensional models with substrate potential. To establish universal characteristic properties of the process, we explore three different models: Frenkel-Kontorova (FK), phi4+ (φ(4)+), and phi4- (φ(4)-). Direct numeric simulations reveal in all these models a crossover from oscillatory decay of short-wave perturbations of the temperature field to smooth diffusive decay of the long-wave perturbations. Such behavior is inconsistent with the parabolic Fourier equation of heat conduction and clearly demonstrates the necessity for hyperbolic corrections in the phenomenological description of the heat conduction process. The crossover wavelength decreases with an increase in the average temperature. The decay patterns of the temperature field almost do not depend on the amplitude of the perturbations, so the use of linear evolution equations for the temperature field is justified. In all models investigated, the relaxation of thermal perturbations is exponential, contrary to a linear chain, where it follows a power law. The most popular lowest-order hyperbolic generalization of the Fourier law, known as the Cattaneo-Vernotte or telegraph equation, is also not valid for the description of the observed behavior of the models with the substrate potential, since the characteristic relaxation time in an oscillatory regime strongly depends on the excitation wavelength. For some of the models, this dependence seems to obey a simple scaling law.

Enhancing surface heat transfer by carbon nanofins: towards an alternative to nanofluids?

BACKGROUND

Nanofluids are suspensions of nanoparticles and fibers which have recently attracted much attention because of their superior thermal properties. Nevertheless, it was proven that, due to modest dispersion of nanoparticles, such high expectations often remain unmet. In this article, by introducing the notion of nanofin, a possible solution is envisioned, where nanostructures with high aspect-ratio are sparsely attached to a solid surface (to avoid a significant disturbance on the fluid dynamic structures), and act as efficient thermal bridges within the boundary layer. As a result, particles are only needed in a small region of the fluid, while dispersion can be controlled in advance through design and manufacturing processes.

RESULTS

Toward the end of implementing the above idea, we focus on single carbon nanotubes to enhance heat transfer between a surface and a fluid in contact with it. First, we investigate the thermal conductivity of the latter nanostructures by means of classical non-equilibrium molecular dynamics simulations. Next, thermal conductance at the interface between a single wall carbon nanotube (nanofin) and water molecules is assessed by means of both steady-state and transient numerical experiments.

CONCLUSIONS

Numerical evidences suggest a pretty favorable thermal boundary conductance (order of 107 W·m-2·K-1) which makes carbon nanotubes potential candidates for constructing nanofinned surfaces.

Scaling and the thermal conductivity of the Frenkel-Kontorova model

We have studied the dependence of the thermal conductivity kappa on the strength of the interparticle potential lambda and the strength of the external potential beta in the Frenkel-Kontorova model. We found that the functional relation can be expressed in a scaling form, kappa proportional, variantlambda;{32}/beta;{2} . This result is first obtained by nonequilibrium molecular dynamics. It is then confirmed by two analytical methods, the self-consistent phonon theory and the self-consistent stochastic reservoirs method. The thermal conductivity kappa is therefore a decreasing functon of beta and an increasing function of lambda.

Symmetry properties of heat conduction in inhomogeneous materials

We address the analysis of the energy flow properties of lattice Hamiltonian systems: precisely, we investigate, by using analytical methods, a schematically anharmonic and inhomogeneous model: namely, the chain of oscillators with self-consistent reservoirs. We obtain a symmetric thermal conductivity even for a system with inhomogeneous interparticle interactions or with graded particle masses. Our results show that inhomogeneity in a system obeying Fourier's law does not guarantee an asymmetric heat flow, and they pose the question if only anharmonicity and inhomogeneity are sufficient to assure a significative thermal rectification in a system, as suggested in the recent literature for graded structures.

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.

Parameter-dependent thermal conductivity of one-dimensional phi4 lattice

We examine the thermal conductivity of a one-dimensional phi4 lattice with strong on-site harmonic potential. The expression for the thermal conductivity in terms of different parameters is derived from the effective phonon theory. Numerical calculations using nonequilibrium molecular dynamics are compared with the predictions of the effective phonon theory and the theory of the Peierls-Boltzmann transport equation. It is found that the numerical results are consistent with the prediction of the effective phonon theory in the intermediate parameter range and approach the predictions of Peierls-Boltzmann transport theory in the strongly pinned limit.

Quantum mechanical heat transport in disordered harmonic chains

We investigate the mechanism of heat conduction in ordered and disordered harmonic one-dimensional chains within the quantum mechanical Langevin method. In the case of disordered chains we find indications for normal heat conduction, which means that there is a finite temperature gradient, but we cannot clearly decide whether the heat resistance increases linearly with the chain length. Furthermore, we observe characteristic quantum mechanical features like the Bose-Einstein statistics of the occupation numbers of the normal modes, freezing of the heat conductivity, and influence of the entanglement within the chain on the current. For the ordered chain we recover some classical results like a vanishing temperature gradient and a heat flux independent of the length of the chain.

Changing the heat conductivity: an analytical study

We address the problem of controlling the energy flow in lattice systems: we investigate, in an analytical approach, the properties of the heat conductivity of the harmonic crystal with self-consistent stochastic reservoirs, a single model with normal conductivity. For the case of weak interparticle interaction, in a perturbative analysis, we obtain an expression for the thermal conductivity and show how to decrease and/or increase the heat current inside the system by changing the masses of the particles and/or the on-site potentials. These results may be useful in the construction of devices controlling the heat conduction.

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.

Interface thermal resistance between dissimilar anharmonic lattices

We study interface thermal resistance (ITR) in a system consisting of two dissimilar anharmonic lattices exemplified by the Fermi-Pasta-Ulam and Frenkel-Kontorova models. It is found that the ITR is asymmetric; namely, it depends on how the temperature gradient is applied. The dependence of the ITR on the coupling constant, temperature, temperature difference, and system size is studied. Possible applications in nanoscale heat management and control are discussed.

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) (0<alpha< or =2), then the thermal conductivity can be expressed in terms of the system size L as kappa=cL(beta) with beta=2-2/alpha. This result predicts that a normal diffusion (alpha=1) implies a normal heat conduction obeying the Fourier law (beta=0), a superdiffusion (alpha>1) 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.

Nonequilibrium statistical mechanics of anharmonic crystals with self-consistent stochastic reservoirs

We consider a d -dimensional crystal with an arbitrary harmonic interaction and an anharmonic on-site potential, with a stochastic Langevin heat bath at each site. We develop an integral formalism for the correlation functions that is suitable for the study of their relaxation (time decay) as well as their behavior in space. Furthermore, in a perturbative analysis, for the one-dimensional system with weak coupling between the sites and small quartic anharmonicity, we investigate the steady state and show that Fourier's law holds. We also obtain an expression for the thermal conductivity (for arbitrary next-neighbor interactions) and give the temperature profile in the steady state.

Heat conduction in a one-dimensional chain of hard disks with substrate potential

Heat conduction in a one-dimensional chain of equivalent rigid particles in the field of the external on-site potential is considered. The zero diameters of the particles correspond to the integrable case with the divergent heat conduction coefficient. By means of a simple analytical model it is demonstrated that for any nonzero particle size the integrability is violated and the heat conduction coefficient converges. The result of the analytical computation is verified by means of numerical simulation in a plausible diapason of parameters, and good agreement is observed.

Fermi-Pasta-Ulam beta lattice: Peierls equation and anomalous heat conductivity

The Peierls equation is considered for the Fermi-Pasta-Ulam beta lattice. Explicit form of the linearized collision operator is obtained. Using this form the decay rate of the normal-mode energy as a function of wave vector k is estimated to be proportional to k(5/3). This leads to the t(-3/5) long-time behavior of the current correlation function, and, therefore, to the divergent coefficient of heat conductivity. These results are in good agreement with the results of recent computer simulations. Compared to the results obtained through the mode coupling theory our estimations give the same k dependence of the decay rate but a different temperature dependence. Using our estimations we argue that adding a harmonic on-site potential to the Fermi-Pasta-Ulam beta lattice may lead to finite heat conductivity in this model.

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 <Deltax(2)>=2Dt(alpha)(0<alpha</=2), then the thermal conductivity can be expressed in terms of the system size L as kappa=cL(beta) with beta=2-2/alpha. This result predicts that normal diffusion (alpha=1) implies normal heat conduction obeying the Fourier law (beta=0) and that superdiffusion (alpha>1) 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.