Heat transfer in infinite harmonic one-dimensional crystals

Кривцов, А. М.

doi:10.1134/s1028335815090062

Cited by 61 publications

(96 citation statements)

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“…The Maxwell-Cattaneo-Vernotte law also fails to describe the heat transport in FPU chains [16,17]. Harmonic approximation allows to derive equations [18,19] and closed-form solutions [20,21] describing heat transport in chains. However the question arises wether temperature field, obtained in harmonic approximation, can be used for estimation of thermoelastic effects (e.g.…”

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confidence: 99%

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Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature

Kuzkin

Кривцов

2020

Phys. Rev. E

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We study conversion of thermal energy to mechanical energy and vice versa in α-FPU chain with spatially sinusoidal profile of initial temperature. We show analytically that thermal expansion and temperature oscillations, caused by quasiballistic heat transport, excite mechanical vibrations with growing amplitude. This new phenomenon is referred to as "ballistic resonance". At large times, the mechanical vibrations decay monotonically. No recurrence, typical for the original FPU problem, is observed. We assume that the absence of recurrence is due to finite initial temperature of the chain.Energy of an isolated solid body can be converted from a mechanical form to a thermal form and vice versa. Conversion of mechanical energy to thermal energy leads, for example, to damping of mechanical vibrations (or waves). Thermal expansion and heat transport cause an opposite process, i.e. conversion of thermal energy to mechanical energy.In macroscopic continuum theories, the conversion is modeled by coupling between the equation of momentum balance and the equation of energy balance. An example of a continuum theory, describing the conversion, is the linear thermoviscoelasticity. Macroscopic theory of linear thermoviscoelasticity is well-developed [1]. However at micro-and nanoscale this theory faces significant difficulties, because macroscopic constitutive relations can be inapplicable at lower length scales. For example, recent theoretical [2-4] and experimental studies [5][6][7][8][9] show that at micro-and nanoscale the Fourier law of heat conduction can be violated. In particular, ballistic regime of heat transport is observed [10,11].A convenient model for investigation of thermomechanical processes at micro-and nanoscale is the Fermi-Pasta-Ulam (FPU) chain [12]. Despite the apparent simplicity of the model, analytical description of thermoelasticity, heat transport and energy conversion in the FPU chain remains a serious challenge for theoreticians.Several anomalies of thermomechanical behavior are observed for the FPU chain. The heat transport in the FPU model is anomalous, i.e. the Fourier law is not satisfied [13][14][15]. The Maxwell-Cattaneo-Vernotte law also fails to describe the heat transport in FPU chains [16,17]. Harmonic approximation allows to derive equations [18,19] and closed-form solutions [20, 21] describing heat transport in chains. However the question arises wether temperature field, obtained in harmonic approximation, can be used for estimation of thermoelastic effects (e.g. excitation of mechanical vibrations due to thermal expansion). We address this question below.Conversion of mechanical energy to thermal energy is even more challenging. Studies of this process have a long history, starting from the pioneering work of Fermi, Pasta, and Ulam [12]. In paper [12], initial conditions, corresponding to the first normal mode of a chain, were considered. It was shown numerically that energy of this normal mode demonstrates almost periodic behavior, i.e. the system does not reach thermal equilibrium...

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“…We assume that mechanical vibrations of the chain are described by the equation of linear thermoelasticity [26], while behavior of temperature is described by the ballistic heat equation [18,19]. Conversion of mechanical energy to thermal energy is neglected.…”

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confidence: 99%

Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature

Kuzkin

Кривцов

2020

Phys. Rev. E

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“…Coming back to 1D model chains, up to now, many ingredients related to anomalous thermal conduction have been carefully considered, among them are chaos [28][29][30][31][32][33], conservation of momentum [34][35][36][37], asymmetric interactions [38][39][40][41][42], linearity [33,[43][44][45][46], integrability [47], pressure [48,49], surface scattering effect [50] etc. Particularly, the viewpoint that a system with (without) a momentum conservation property should disobey (obey) Fourier's law has been widely accepted, although there are still some contradictory results on this issue [51][52][53][54][55][56][57][58][59].…”

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confidence: 99%

Crossover from ballistic to normal heat transport in theϕ4lattice: If nonconservation of momentum is the reason, what is the mechanism?

2017

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Anomalous (non-Fourier's) heat transport is no longer just a theoretical issue since it has been observed experimentally in a number of low-dimensional nanomaterials, such as SiGe nanowires, carbon nanotubes, and others. To understand these anomalous behaviors, exploring the microscopic origin of normal (Fourier's) heat transport is a fascinating theoretical topic. However, this issue has not yet been fully understood even for one-dimensional (1D) model chains, in spite of a great amount of thorough studies done to date. From those studies it has been widely accepted that the conservation of momentum is a key ingredient to induce anomalous heat transport, while momentumnonconserving systems usually support normal heat transport where Fourier's law is valid. But if the nonconservation of momentum is the reason, what is the underlying microscopic mechanism for the observed normal heat transport? Here we carefully revisit a typical 1D momentum-nonconserving φ 4 model and present evidence that the mobile discrete breathers or, in other words, the moving intrinsic localized modes with frequency components above the linear phonon band can be responsible for that.

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“…Considerable progress has been achieved for harmonic crystals. [19][20][21]24,25,27,28] An analytical description of the ballistic heat transfer in harmonic lattices has been developed in [22,23]. For a one-dimensional nonquantum case, it allows us to obtain the macroscopic heat conduction equation and the corresponding anomalous heat conduction law (alternative to the Fourier law).…”

Section: Introductionmentioning

confidence: 99%

Thermal processes in a one‐dimensional crystal with regard for the second neighbor interaction

et al. 2019

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An influence of the second neighbor interaction on the process of heat propagation in a one‐dimensional crystal is studied. Previously developed model of the ballistic nature of the heat transfer is used. It is shown that the initial thermal perturbation evolves into two consecutive thermal wave fronts propagating with finite and different velocities. The velocity of the first front corresponds to the maximum group velocity of the discrete crystalline model. The velocity of the second front is determined by the second group‐velocity extremum, which arises at a certain ratio between the stiffnesses of the first and second neighbor interaction in the lattice.

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Heat transfer in infinite harmonic one-dimensional crystals

Cited by 61 publications

References 10 publications

Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature

Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature

Crossover from ballistic to normal heat transport in theϕ4lattice: If nonconservation of momentum is the reason, what is the mechanism?

Thermal processes in a one‐dimensional crystal with regard for the second neighbor interaction

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