Viscous heat backflow and temperature resonances in extreme thermal conductors

TL;DR

This work demonstrates that viscous heat hydrodynamics can be robustly induced, controlled, and amplified in extreme thermal conductors such as graphite and layered hBN by solving viscous heat equations (VHE) derived from first-principles LBTE. It reveals steady-state heat backflow and temperature inversion arising from finite thermal viscosity, and shows transient temperature waves with resonant amplification in devices engineered with tunnel-chamber geometries and boundary conditions. By comparing VHE, DPLE, Fourier theory, and full LBTE simulations, the study clarifies the distinct roles of viscosity, boundary scattering, and device geometry in heat transport, and provides concrete strategies to harness hydrodynamic heat flow in next-generation electronics and phononics. The results establish quantitative, parameter-driven predictions for material systems and geometries, enabling experimental observation and practical exploitation of viscous heat backflow and temperature resonances. This work thus opens pathways toward thermal signal processing and neuromorphic concepts based on heat, with implications for advanced thermal management and phononic technologies.

Abstract

We demonstrate that non-diffusive, fluid-like heat transport, such as heat backflowing from cooler to warmer regions, can be induced, controlled, and amplified in extreme thermal conductors such as graphite and hexagonal boron nitride. We employ the viscous heat equations, i.e., the thermal counterpart of the Navier-Stokes equations in the laminar regime, to show with first-principles quantitative accuracy that a finite thermal viscosity yields steady-state heat vortices, and governs the magnitude of transient temperature waves. Finally, we devise strategies that exploit devices' boundaries and resonance to amplify and control heat hydrodynamics, paving the way for novel experiments and applications in next-generation electronic and phononic technologies.

Figures (23)

• Figure 1: Viscous heat backflow and temperature inversion. In-plane ($x{-}y$) heat flow (streamlines) and temperature (colormap) in a tunnel-chamber device made of graphite. Panel a (b) shows the solution of Fourier's equation (VHE) in the presence of a temperature gradient applied to the tunnel ($T=70{\pm}12.5$ K at $y{=}{\mp} 1.25\mu m$); the other boundaries are adiabatic, i.e. $\nabla T{\cdot} \bm{\hat{n}}{=}0$ (and in VHE also $\bm{u}{\cdot} \bm{\hat{n}}{=}0$), where $\bm{\hat{n}}$ is the unit vector normal to the boundary. In the VHE case we also impose finite-slip boundary conditions with slip length 0.4 $\mu m$bocquetFlowBoundaryConditions2007, corresponding to diffusive phonon-boundary scattering (see end notes for details). In Fourier's case (a), the direction of the temperature gradient in the chamber mirrors that in the tunnel. In contrast, the VHE (b) allows for the emergence of viscous backflow, whose hallmark is a temperature gradient in the chamber reversed compared to the tunnel. Panel c, vorticity of the VHE heat flux, $\nabla {\times}\bm{Q}^{\rm TOT}$; the vorticity for Fourier's flux (not reported) is trivially zero. The density of streamlines is proportional to the magnitude of the heat flux: in Fourier (VHE) case the grey streamlines in the chamber are 100 (10) times closer to each other compared to black streamlines in the tunnel.
• Figure 2: Transient viscous heat backflow. Columns show temperature (left), temperature-gradient heat flux ($\bm{Q}^{\delta}$, center), and drifting heat flux ($\bm{Q}^{D}$, right) at different times (relaxation starts at $t{=}0.4$ ns, see text), and are obtained solving the VHE in a graphitic device thermalised at 80 K in the shaded boundary regions. In the 2D plots, color is the heat-flux modulus, and white streamlines show its direction.
• Figure 3: Resonant amplification of temperature waves in graphite at natural isotopic disorder (98.9% $^{12}C$, 1.1% $^{13}C$) and around $T{=}80 K$ (a); the prediction from the viscous VHE is green, red is Fourier's law, and the DPLE is reported in inset (b). Inset (c), the VHE predict that isotopically purified graphite (99.9% $^{12}C$, 0.1% $^{13}C$, dashed dark green) features a stronger resonant amplification compared to natural graphite (solid green); in addition, analogous signatures are predicted to appear in h$^{11}$BN, around $T{=}60 K$ in 15$\mu m$-long devices.
• Figure 4: Maximum resonant amplification as a function of size, temperature, and isotopic disorder. The colormaps show the maximum resonant amplification (see text) as a function of the characteristic size of the rectangular device footnote_legend and of the temperature around which the perturbation is applied. Panels a,b) refer to natural graphite, a) is the DPLE and b) the VHE. Empty green circles have an area proportional to the hydrodynamic strength footnote_fig4 measured in experiments by Huberman2019 for natural samples, and by Ding2022 for isotopically purified samples; the red crosses correspond to diffusive behavior (no hydrodynamics). The VHE (b) capture the temperatures and lengthscales at which hydrodynamic behavior appears; in contrast, the inviscid DPLE departs from experiments at low temperature. Panel c), VHE hydrodynamic behavior in isotopically purified graphite, in broad agreement with experiments Ding2022. Panel d), VHE resonant behavior in h$^{11}$BN.
• Figure 5: Viscous heat backflow and temperature inversion from VHE & LBTE, and dependence on BCs. Top row: in-plane ($x{-}y$) heat flow (streamlines) and temperature (colormap); bottom row: vorticity. All panels show a tunnel-chamber device made of natural graphite having a temperature difference applied to the tunnel (i.e., tunnel's boundaries at $y{=}{\mp} 1.25\mu m$ are thermalized at $70\pm 12.5$ K). The non-thermalized boundaries are always adiabatic---i.e., $\bm{Q}^{TOT}\cdot \bm{\hat{n}}=0$, where $\bm{\hat{n}}$ is the unit vector orthogonal to the boundary---and each column shows the solution of VHE or LBTE for BCs exerting different friction on the total heat-flux component tangential to the boundary: a, idealized no-slip BCs $\bm{u}=0$ within VHE; b, more realistic, 0.4 $\mu m$ finite-slip-length BCs within mesoscopic VHE, corresponding to microscopic diffusive phonon-boundary scattering raya-moreno_bte-barna_2022 within LBTE (c, see text); d and e are idealized, frictionless boundaries, corresponding to zero-shear-stress BCs within VHE (d) and reflective phonon-boundary scattering raya-moreno_heat_nodate within LBTE (e). The thistle and beige circle of diameter 1 $\mu m$ in the top and bottom parts of the chamber (centered at $x=1.02\mu m, \;y=\pm 0.58\mu m$), respectively, are temperature probes that contain the local maxima (thistle) and minima (beige) of the temperature profile, highlighting that the emergence of viscous backflow implies a temperature gradient in the chamber inverted compared to the tunnel, with inversion being stronger as the boundaries' friction is reduced from left to right. We highlight how VHE and LBTE yield a heat flux with closed streamlines, a sufficient condition to have nonzero vorticity, as confirmed by the bottom row.