Crystals and glasses exhibit fundamentally different heat conduction mechanisms: the periodicity of crystals allows for the excitation of propagating vibrational waves that carry heat, as first discussed by Peierls 1 ; in glasses, the lack of periodicity breaks Peierls' picture and heat is mainly carried by the coupling of vibrational modes, often described by a harmonic theory introduced by Allen and Feldman 2 . Anharmonicity or disorder are thus the limiting factors for thermal conductivity in crystals or glasses; hitherto, no transport equation has been able to account for both. Here, we derive such equation, resulting in a thermal conductivity that reduces to the Peierls and Allen-Feldman limits, respectively, in anharmonicand-ordered or harmonic-and-disordered solids, while also covering the intermediate regimes where both effects are relevant. This approach also solves the longstanding problem of accurately predicting the thermal properties of crystals with ultralow or glass-like thermal conductivity 3-10 , as we show with an application to a thermoelectric material representative of this class.In 1929 Peierls 1 developed a semi-classical theory for heat conduction in terms of a Boltzmann transport equation (BTE) for propagating phonon wave packets. Nowadays, modern algorithms and computing systems allow to solve its linearized form (LBTE) either approximately (in the so-called single mode approximation (SMA) 11 ) or exactly, using iterative 12,13 , variational 14 , or exact diagonalization 15,16 methods; its accuracy has been highlighted in many studies [14][15][16][17] . Nevertheless, these cases are characterized by having few, well-separated phonon branches and anharmonicity-limited thermal conductivity; we will refer to these in the following as "simple" crystals. In 1963 Hardy was able to express the thermal conductivity in terms of the phonon velocity operator 18 and showed that its diagonal elements are the phonon group velocities entering the Peierls' BTE, while the off-diagonal terms, missing from it, are actually negligible in simple crystals 18 . In 1989 Allen and Feldman 2 envisioned that these off-diagonal elements, neglected so far, could become dominant in disordered regimes, where Peierls' picture breaks down due to the impossibility of defining phonons and group velocities. As a consequence, a harmonic theory of thermal transport in glasses was introduced, where disorder limits thermal conductivity and heat is carried by couplings of vibrational modes arising from the off-diagonal elements of the velocity operator (diffusons and locons 19,20 ). Recently, it has been argued that the diffuson conduction mechanism, typical of glasses, can also emerge in a third class of materials, termed "complex" crystals 7 , characterized by large unit cells and many quasi-degenerate phonon branches, where it coexists with phonon transport. Conversely, crystal-like prop-agation mechanisms have been suggested also for glasses (propagons 19 ) -albeit without a formal justification -in order to explain the ex...

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Heat conduction in dielectric crystals originates from the propagation of atomic vibrational waves, whose microscopic dynamics is well described by linearized or generalized phonon Boltzmann transport. Recently, it was shown that the thermal conductivity can be resolved exactly and in a closed form as a sum over relaxons, i.e. the collective phonon excitations that are eigenvectors of Boltzmann equation's scattering matrix [Cepellotti and Marzari, Phys. Rev. X 6, 041013 (2016)]. Relaxons have a well-defined parity and only odd relaxons contribute to the thermal conductivity. Here, we show that the complementary set of even relaxons determines another quantity -the thermal viscosity -that enters into the description of heat transport in the hydrodynamic regime, where dissipation of crystal momentum by Umklapp scattering phases out. We also show how the thermal viscosity and conductivity parametrize two novel viscous heat equations -two coupled equations for the local temperature and drift velocity fields -which represent the thermal counterpart of the Navier-Stokes equations of hydrodynamics in the linear, laminar regime. These viscous heat equations are derived from a coarse-graining of the linearized Boltzmann transport equation for phonons, and encompass both limits of Fourier's law or second sound for strong or weak Umklapp dissipation, respectively. Last, we introduce the Fourier deviation number, a dimensionless parameter that quantifies the steady-state deviations from Fourier's law due to hydrodynamic effects. We showcase these findings in a test case of a complex-shaped device made of silicon or diamond. This formulation generalizes rigorously Fourier's heat equation, and extends the reach of microscopic computational techniques to characterize the fundamental parameters governing heat conduction. * michele.simoncelli@epfl.ch ics. Sussmann and Thellung [12], starting from the linearized BTE (LBTE) in absence of momentumdissipating (Umklapp) phonon-phonon scattering events, derived mesoscopic equations in terms of temperature and phonon drift velocity, the thermal counterpart of pressure and fluid velocity in liquids. Further advances came from Gurzhi [13,14] and Guyer & Krumhansl [15,16] who, including the effect of weak crystal momentum dissipation, obtained equations for damped second sound and Poiseuille heat flow. Among early works, we also mention the discussions on phonon hydrodynamics using approaches different from the LBTE of Refs. [17,18]. While correctly capturing the qualitative features of phonon hydrodynamics, all the theoretical investigations mentioned above are heuristic, e.g. they assume simplified phonon dispersion relations (either power-law [13,14] or linear isotropic [12,15,16]), or neglect momentum dissipation. A more rigorous and general formulation -albeit valid only in the hydrodynamic regime of weak Umklapp scattering -was introduced by Hardy, who extended the discussion of second sound [19] and, together with Albers, of Poiseuille flow in terms of mesoscopic transport equations...

Capacitive mixing (CapMix) and capacitive deionization (CDI) are currently developed as alternatives to membrane-based processes to harvest blue energy-from salinity gradients between river and sea waterand to desalinate water-using charge-discharge cycles of capacitors. Nanoporous electrodes increase the contact area with the electrolyte and hence, in principle, also the performance of the process. However, models to design and optimize devices should be used with caution when the size of the pores becomes comparable to that of ions and water molecules. Here, we address this issue by simulating realistic capacitors based on aqueous electrolytes and nanoporous carbide-derived carbon (CDC) electrodes, accounting for both their complex structure and their polarization by the electrolyte under applied voltage. We compute the capacitance for two salt concentrations and validate our simulations by comparison with cyclic voltammetry experiments. We discuss the predictions of Debye-Hückel and Poisson-Boltzmann theories, as well as modified Donnan models, and we show that the latter can be parametrized using the molecular simulation results at high concentration. This then allows us to extrapolate the capacitance and salt adsorption capacity at lower concentrations, which cannot be simulated, finding a reasonable agreement with the experimental capacitance. We analyze the solvation of ions and their confinement within the electrodes-microscopic properties that are much more difficult to obtain experimentally than the electrochemical response but very important to understand the mechanisms at play. We finally discuss the implications of our findings for CapMix and CDI, both from the modeling point of view and from the use of CDCs in these contexts.