Heat transfer in a 1D harmonic crystal

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Virtual laboratory > Heat transfer in a 1D harmonic crystal
A.M. Krivtsov (analytical silution, simulation algorithms), D.V. Tsvetkov (programming, calculation algorithms).

Heat transfer in the simplest discrete systems doesn’t obey common macroscopic laws. Recently experimentalists have observed the similar behavior at nanolevel, in molecular and atomic systems. The simulation below demonstrates heat transfer process in a 1D harmonic crystal. Two graphs are shown: results of molecular dynamics simulation and corresponding continuum solution. You can also compare the results with predictions of other continuum models. The analysis of the system and derivation of the continuum solution are presented in paper: ArXiv:1509.02506 (abstract, pdf). See also information below the simulation.

Use the Restart button to see the process from the beginning.

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Discrete model (microlevel)

We consider a one-dimensional crystal, described by the following equations of motion:

[math] \ddot{u}_i = \omega_0^2(u_{i-1}-2u_i+u_{i+1}) ,\qquad \omega_0 = \sqrt{C/m}, [/math]

where [math]u_i[/math] is the displacement of the [math]i[/math]th particle, [math]m[/math] is the particle mass, [math]C[/math] is the stiffness of the interparticle bond. The crystal is infinite: the index [math]i[/math] is an arbitrary integer. The initial conditions are

[math] u_i|_{t=0} = 0 ,\qquad \dot u_i|_{t=0} = \sigma(x)\varrho_i , [/math]

where [math]\varrho_i[/math] are independent random values with zero expectation and unit variance; [math]\sigma^2(x)[/math] is variance of the initial velocities, which is a slowly varying function of the spatial coordinate [math]x=ia[/math], where [math]a[/math] is the lattice constant. These initial conditions correspond to an instantaneous temperature perturbation, which can be induced in crystals, for example, by an ultrashort laser pulse. Periodic conditions are used at the boundaries.

Kinetic temperature (link between micro and macro)

The kinetic temperature [math]T[/math] is defined as

[math] T(x) = \frac m{k_{B}}\langle\dot u_i^2\rangle, [/math]

where [math]k_{B}[/math] is the Boltzmann constant, [math]i=x/a[/math], angle brackets stand for mathematical expectation.

Continuum description (macrolevel)

Reversible heat wave equation: [math]\ddot T +\frac1t\dot T = c^2 T''[/math] — this equation is derived as a direct consequence of the discrete microscopic equations [1]

Notations: [math]t[/math] is time (variable), [math]c[/math] is the sound speed.

Classic continuum equations

Heat (Fourier): [math]\dot T = \beta T''[/math] [2]

Heat wave (MCV): [math]\ddot T +\frac1\tau\dot T = \frac\beta\tau T''[/math]

Wave (d’Alembert): [math]\ddot T = c^2 T''[/math] [3]

Notations: [math]\tau[/math] is the relaxation time (constant), [math]\beta[/math] is the thermal diffusivity, [math]\kappa[/math] is the thermal conductivity, [math]\rho[/math] is the density, MCV stands for Maxwell-Cattaneo-Vernotte.

Related publications

  • A.M. Krivtsov. Energy oscillations in a one-dimensional crystal. Doklady Physics, 2014, Vol. 59, No. 9, pp. 427–430 (pdf: 162 Kb)