Abstract
In the paper, we deal with ballistic heat transport in a graphene lattice subjected to a point heat source. It is assumed that a graphene sheet is suspended under tension in a viscous gas. We use the model of a harmonic polyatomic (more exactly diatomic) lattice performing out-of-plane motions. The dynamics of the lattice is described by an infinite system of stochastic ordinary differential equations with white noise in the right-hand side, which models the point heat source. On the basis of the previous analytical unsteady analysis, an analytical formula in continuum approximation is suggested, which allows one to describe a steady-state kinetic temperature distribution in the graphene lattice in continuum approximation. The obtained solution is in a good agreement with numerical results obtained for the discrete system everywhere excepting a neighbourhood of six singular rays with the origin at the heat source location. The continuum solution becomes singular at these rays, unlike the discrete one, which appears to be localized in a certain sense along the rays. The factors, which cause such a directional localization and the mismatch between the continuum and discrete solutions, are discussed. We expect that the suggested formula is applicable for various damped polyatomic lattices where all particles have equal masses in the case of universal for all particles external viscosity.
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Notes
To describe such sources, one needs to formulate a problem for a system of stochastic ordinary differential equations.
\(a\simeq 0.142\) nm for real graphene lattice [35].
Everywhere, excluding the singular rays in the case when the steady-state solution is considered, see Sect. 6.
Kuzkin actually considered a more general problem formulation: the external random excitation is not assumed to be equal for all particles in a cell (Eq. (3.16) generally is not assumed to be true), and the masses of the particles inside a cell are also not assumed to be equal.
Note that in the case of a primitive rhombic scalar lattice (in particular for a square lattice) the singular rays also exist and correspond to \(\alpha \in \left\{ -\frac{\pi }{2},0,\frac{\pi }{2}\right\} =0\), see [14].
Excluding the points, where \({{\mathbf {g}}}_\pm =\mathbf{0}\) and the corresponding direction is undefined, which are shown as the green boxes.
The non-stationary solution is almost vanishes outside the circle with radius \(g_{\mathrm {max}}t\) with centre at the point source location, where \(g_{\mathrm {max}}\simeq 0.897\omega _*a\) [30] is the maximum magnitude for the vectors \({{\mathbf {g}}}_\pm \).
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Acknowledgements
The authors are grateful to A. T. Ivaschenko, V. A. Kuzkin, O. V. Gendelman, A. Politi, E. V. Shishkina, A. A. Sokolov for useful and stimulating discussions.
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Appendix A: The dispersion surfaces and the group velocities for graphene lattice
Appendix A: The dispersion surfaces and the group velocities for graphene lattice
Consider Eqs. (3.6) and (3.7) in the absence of dissipation (\(\eta =0\)) and of the noise term in the right-hand side (\(b_0=0\)):
Let
Provided that \(\omega \) satisfies the dispersion relations
the right-hand side of Eq. (A.2) is the solution of Eq. (A.1). Here
is the wave vector, \(\mathbf{b}^\gamma \) is the dual basis defined by Eq. (4.9).
The dispersion relations for graphene lattice are found in [1, 22, 29]:
The plot of the dispersion surfaces is shown in Fig. 9.
The vectors of the group velocities are
Here
is a characteristic speed.
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Gavrilov, S.N., Krivtsov, A.M. Steady-state ballistic thermal transport associated with transversal motions in a damped graphene lattice subjected to a point heat source. Continuum Mech. Thermodyn. 34, 297–319 (2022). https://doi.org/10.1007/s00161-021-01059-3
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DOI: https://doi.org/10.1007/s00161-021-01059-3