Heat transfer in a 1D harmonic crystal

Virtual laboratory > Heat transfer in a 1D harmonic crystal

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A.M. Krivtsov (analytical silution, simulation algorithms), D.V. Tsvetkov (programming, calculation algorithms).

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Heat transfer in the simplest discrete systems doesn’t obey the known 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: A.M. Krivtsov, On unsteady heat conduction in a harmonic crystal. ArXiv:1509.02506 (abstract, pdf).

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

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[/math] is variance of the initial velocities of the particles, 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.

Macroscopic equations

— Reversible (Krivtsov): [math]\ddot T +\frac1t\dot T = c^2 T''[/math] [1]

Macroscopic 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]t[/math] is time (variable), [math]\tau[/math] is the relaxation time (constant), [math]\beta[/math] is the thermal diffusivity, [math]\kappa[/math] is the thermal conductivity, [math]c[/math] is the sound speed, [math]\rho[/math] is the density, MCV stands for Maxwell-Cattaneo-Vernotte.

Related publications

• A.M. Krivtsov. On unsteady heat conduction in a harmonic crystal. ArXiv:1509.02506 (abstract, pdf, simulation)

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