Steady-state ballistic thermal transport associated with transversal motions in a damped graphene lattice subjected to a point heat source

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.

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 ($b_0=0$):

$$\begin{aligned} m\partial _t^2{{\varvec{u}}}(\breve{\mathbf{x}}) +{\varvec{C}}_0 {\varvec{u}}(\breve{\mathbf{x}}) +\sum _{i=1}^{Q/2} \left( {\varvec{C}}_1 {\varvec{u}}(\breve{\mathbf{x}}+\mathbf{b}_i) +{\varvec{C}}_1^\top {\varvec{u}}(\breve{\mathbf{x}}-\mathbf{b}_i) \right) = \mathbf{0}. \end{aligned}$$

(A.1)

Let

$$\begin{aligned} {\varvec{u}}(\breve{\mathbf{x}})={\varvec{U}}\exp (-\mathrm {i}\mathbf{p}\cdot \breve{\mathbf{x}}-\mathrm {i}\omega t). \end{aligned}$$

(A.2)

Provided that $\omega$ satisfies the dispersion relations

$$\begin{aligned} \omega =\omega _\pm (p_1,p_2), \end{aligned}$$

(A.3)

the right-hand side of Eq. (A.2) is the solution of Eq. (A.1). Here

$$\begin{aligned} \mathbf{p}=p_\gamma \mathbf{b}^\gamma \end{aligned}$$

(A.4)

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]:

$$\begin{aligned}&\omega _\pm ^2=\omega _*^2(3\pm R(p_1,p_2)), \end{aligned}$$

(A.5)

$$\begin{aligned}&\quad R(p_1,p_2)=\sqrt{3+2(\cos p_1 +\cos p_2 +\cos (p_1-p_2) )}, \end{aligned}$$

(A.6)

$$\begin{aligned}&\quad \omega _*^2=\frac{C}{m}. \end{aligned}$$

(A.7)

The plot of the dispersion surfaces is shown in Fig. 9.

Fig. 9

Acoustic $\omega _-(p_1,p_2)$ and optic $\omega _+(p_1,p_2)$ dispersion surfaces ($\omega _*=1$)

The vectors of the group velocities are

$$\begin{aligned}&\displaystyle {{{\mathbf {g}}}}_\pm = \frac{\partial \omega _\pm }{\partial \mathbf{p}} = \frac{\partial \omega _\pm {}}{\partial p_\gamma }\mathbf{b}_\gamma , \end{aligned}$$

(A.8)

$$\begin{aligned}&\displaystyle \quad g^1_\pm =\frac{\partial \omega _\pm }{\partial p_1} =\mp \frac{\omega _*^2(\sin p_1+\sin (p_1-p_2))}{2\omega _\pm R}, \end{aligned}$$

(A.9)

$$\begin{aligned}&\displaystyle \quad g^2_\pm =\frac{\partial \omega _\pm }{\partial p_2} =\mp \frac{\omega _*^2(\sin p_2-\sin (p_1-p_2))}{2\omega _\pm R}, \end{aligned}$$

(A.10)

$$\begin{aligned}&\displaystyle \quad {{\mathbf {g}}}_\pm =\mp \frac{\sqrt{3}c\big ( (2\sin (p_1-p_2)+\sin p_1-\sin p_2)\mathbf{e}_1 +\sqrt{3}(\sin p_1+\sin p_2)\mathbf{e}_2 \big )}{4R\sqrt{3\pm R}}. \end{aligned}$$

(A.11)

Here

$$\begin{aligned} c=\omega _*a \end{aligned}$$

(A.12)

is a characteristic speed.