Dimensional crossover of heat conduction in low dimensions

Heat transport in an angular-momentum-conserving lattice

It is expected that the energy-diffusion propagator in a one-dimensional nonlinear lattice with three conserved quantities: energy, momentum, and stretch, consists of a central heat mode and two sound modes. The heat mode follows a Lévy distribution. Consequently, the heat diffusion is super, i.e., the second moment of the diffusion propagator diverges as t^{β} with β>1; and the heat conduction is anomalous, i.e., the heat conductivity is size dependent and diverges with size N by N^{α}, with α>0. In this paper, we study a one-dimensional lattice with two-dimensional transverse motions, in which the total angular momentum also conserves. More importantly, the diffusion of this conserved quantity is ballistic. Surprisingly, the above pictures and the values of the mentioned power exponents keep unchanged. The universality of the scalings is then further extended. On the other hand, the detailed strengths of heat transports are largely enhanced. Such a counterintuitive finding can be explained by the change of the phonon mean-free path of the lattices.

Energy diffusion in two-dimensional momentum-conserving nonlinear lattices: Lévy walk and renormalized phonon

The energy diffusion process in a few two-dimensional Fermi-Pasta-Ulam-type lattices is numerically simulated via the equilibrium local energy spatiotemporal correlation. Just as the nonlinear fluctuating hydrodynamic theory suggested, the diffusion propagator consists of a bell-shaped central heat mode and a sound mode extending with a constant speed. The profiles of the heat and sound modes satisfy the scaling properties from a random-walk-with-velocity-fluctuation process very well. An effective phonon approach is proposed, which expects the frequencies of renormalized phonons as well as the sound speed with quite good accuracy. Since many existing analytical and numerical studies indicate that heat conduction in such two-dimensional momentum-conserving lattices is divergent and the thermal conductivity κ increases logarithmically with lattice length, it is expected that the mean-square displacement of energy diffusion grows as tlnt. Discrepancies, however, are noticeably observed.

Heat current flows across an interface in two-dimensional lattices

Heat current J that flows through a few typical two-dimensional nonlinear lattices is systematically studied. Each lattice consists of two identical segments that are coupled by an interface with strength k_{int}. It is found that the two-universality-class scenario that is revealed in one-dimensional systems is still valid in the two-dimensional systems. Namely, J may follow k_{int} in two entirely different ways, depending on whether or not the interface potential energy decays to zero. We also study the dependence of J on lattice width N_{Y} and transverse interaction strength k_{Y}. Universal power-law decay or divergence is observed. Finally, we check the equipartition theorem in the systems since it is the basis of all our theoretical analyses. Surprisingly, it holds perfectly even at the interface where there is a finite temperature jump, which makes the system far from equilibrium. However, the equipartition of potential energy, which is observed in one-dimensional systems, is no longer satisfied due to the interaction between different dimensions.

Heat conduction in a three-dimensional momentum-conserving fluid

Size dependence of energy transport and the effects of reduced dimensionality on transport coefficients are of key importance for understanding nonequilibrium properties of matter on the nanoscale. Here, we perform nonequilibrium and equilibrium simulations of heat conduction in a three-dimensional (3D) fluid with the multiparticle collision dynamics, interacting with two thermal walls. We find that the bulk 3D momentum-conserving fluid has a finite nondiverging thermal conductivity. However, for large aspect ratios of the simulation box, a crossover from 3D to one-dimensional (1D) abnormal behavior of the thermal conductivity occurs. In this case, we demonstrate a transition from normal to abnormal transport by a suitable decomposition of the energy current. These results not only provide a direct verification of Fourier's law, but also further confirm the validity of existing theories for 3D fluids. Moreover, they indicate that abnormal heat transport persists also for almost 1D fluids over a large range of sizes.

Heat conduction in two-dimensional momentum-conserving and -nonconserving gases

Compared to that for two-dimensional (2D) lattices, our understanding of heat conduction in 2D gases is still limited. Here we study heat conduction behavior of 2D gas systems with momentum-conserving and -nonconserving interparticle interactions by using the nonequilibrium and equilibrium molecular dynamics methods. For the momentum-conserving system, we find that when the dimensionality of the system is changed from 2D to quasi-one-dimensional (quasi-1D), the heat conductivity κ diverges with the system size L as κ∼lnL (the theoretical prediction for 2D systems) for a short L and shows, in the thermodynamic limit, a tendency to κ∼L^{1/3} like that predicted in 1D fluids. This suggests that the dimensional-crossover effect of heat conduction exists in 2D systems with conserved momentum. In contrast, for the momentum-nonconserving system, as L increases, finite heat conductivity independent of L is observed. These findings are in agreement with the predictions given by hydrodynamic theory and thus further confirm the validity of the theory in 2D gases.

Thermal transport of carbon nanomaterials

The diversity of thermal transport properties in carbon nanomaterials enables them to be used in different thermal fields such as heat dissipation, thermal management, and thermoelectric conversion. In the past two decades, much effort has been devoted to study the thermal conductivities of different carbon nanomaterials. In this review, different theoretical methods and experimental techniques for investigating thermal transport in nanosystems are first summarized. Then, the thermal transport properties of various pure carbon nanomaterials including 1D carbon nanotubes, 2D graphene, 3D carbon foam, are reviewed in details and the associated underlying physical mechanisms are presented. Meanwhile, we discuss several important influences on the thermal conductivities of carbon nanomaterials, including size, structural defects, chemisorption and strain. Moreover, we introduce different nanostructuring pathways to manipulate the thermal conductivities of carbon-based nanocomposites and focus on the wave nature of phonons for controlling thermal transport. At last, we briefly review the potential applications of carbon nanomaterials in the fields of thermal devices and thermoelectric conversion.

Anomalous energy diffusion in two-dimensional nonlinear lattices

Heat transport in one-dimensional (1D) momentum-conserved lattices is generally assumed to be anomalous, thus yielding a power-law divergence of thermal conductivity with system length. However, whether heat transport in a two-dimensional (2D) system is anomalous or not is still up for debate because of the difficulties involved in experimental measurements or due to the insufficiently large simulation cell size. Here we simulate energy and momentum diffusion in the 2D nonlinear lattices using the method of fluctuation correlation functions. Our simulations confirm that energy diffusion in the 2D momentum-conserved lattices is anomalous and can be well described by the Lévy-stable distribution. As is expected, we verify that 2D nonlinear lattices with on-site potentials exhibit normal energy diffusion, independent of the dimension. Contrary to the hypothesis of a 1D system, we further clarify that anomalous heat transport in the 2D momentum-conserved system cannot be corroborated by the momentum superdiffusion any longer. Our findings offer some valuable insights into mechanisms of thermal transport in 2D system.

Delocalization and heat transport in multidimensional trapped ion systems

We study the connection between heat transport properties of systems coupled to different thermal baths in two separate regions and their underlying nonequilibrium dynamics. We consider classical systems of interacting particles that may exhibit a certain degree of delocalization and whose effective dimensionality can be modified through the controlled variation of a global trapping potential. We focus on Coulomb crystals of trapped ions, which offer a versatile playground to shed light on the role that spatial constraints play on heat transport. We use a three-dimensional model to simulate the trapped ion system and show in a numerically rigorous manner to what extent heat transport properties could be feasibly tuned across the structural phase transitions among the linear, planar zigzag, and helical configurations. By solving the classical Langevin equations of motion, we analyze the steady state spatial distributions of the particles, the temperature profiles, and total heat flux through the various structural phase transitions that the system may experience. The results evidence a clear correlation between the degree of delocalization of the internal ions and the emergence of a nonzero gradient in the temperature profiles. The signatures of the phase transitions in the total heat flux as well as the optimal spatial configuration for heat transport are shown.

Divergent and Ultrahigh Thermal Conductivity in Millimeter-Long Nanotubes

Low-dimensional materials could display anomalous thermal conduction that the thermal conductivity (κ) diverges with increasing lengths, in ways inconceivable in any bulk materials. However, previous theoretical or experimental investigations were plagued with many finite-size effects, rendering the results either indirect or inconclusive. Indeed, investigations on the anomalous thermal conduction must demand the sample length to be sufficiently long so that the phenomena could emerge from unwanted finite-size effects. Here we report experimental observations that the κ's of single-wall carbon nanotubes continuously increase with their lengths over 1 mm, reaching at least 8640  W/mK at room temperature. Remarkably, the anomalous thermal conduction persists even with the presence of defects, isotopic disorders, impurities, and surface absorbates. Thus, we demonstrate that the anomalous thermal conduction in real materials can persist over much longer distances than previously thought. The finding would open new regimes for wave engineering of heat as well as manipulating phonons at macroscopic scales.

Phonon thermal conduction in novel 2D materials

Recently, there has been increasing interest in phonon thermal transport in low-dimensional materials, due to the crucial importance of dissipating and managing heat in micro- and nano-electronic devices. Significant progress has been achieved for one-dimensional (1D) systems, both theoretically and experimentally. However, the study of heat conduction in two-dimensional (2D) systems is still in its infancy due to the limited availability of 2D materials and the technical challenges of fabricating suspended samples that are suitable for thermal measurements. In this review, we outline different experimental techniques and theoretical approaches for phonon thermal transport in 2D materials, discuss the problems and challenges of phonon thermal transport measurements and provide a comparison between existing experimental data. Special attention will be given to the effects of size, dimensionality, anisotropy and mode contributions in novel 2D systems, including graphene, boron nitride, MoS₂, black phosphorous and silicene.

Stretch diffusion and heat conduction in one-dimensional nonlinear lattices

For heat conduction in one-dimensional (1D) nonlinear Hamiltonian lattices, it has been known that conserved quantities play an important role in determining the actual heat conduction behavior. In closed or microcanonical Hamiltonian systems, the total energy and stretch are always conserved. Depending on the existence of external on-site potential, the total momentum can be conserved or not. All the momentum-conserving lattices have anomalous heat conduction except the 1D coupled rotator lattice. It was recently claimed that "whenever stretch (momentum) is not conserved in a 1D model, the momentum (stretch) and energy fields exhibit normal diffusion." The stretch in a coupled rotator lattice was also argued to be nonconserved due to the requirement of a finite partition function, which enables the coupled rotator lattice to fulfill this claim. In this work, we will systematically investigate stretch diffusion and heat conduction in terms of energy diffusion for typical 1D nonlinear lattices. Contrary to what was claimed, no clear connection between conserved quantities and heat conduction can be established. The actual situation might be more complicated than what was proposed.

Thermal conductivity of graphene and graphite: collective excitations and mean free paths

We characterize the thermal conductivity of graphite, monolayer graphene, graphane, fluorographane, and bilayer graphene, solving exactly the Boltzmann transport equation for phonons, with phonon-phonon collision rates obtained from density functional perturbation theory. For graphite, the results are found to be in excellent agreement with experiments; notably, the thermal conductivity is 1 order of magnitude larger than what found by solving the Boltzmann equation in the single mode approximation, commonly used to describe heat transport. For graphene, we point out that a meaningful value of intrinsic thermal conductivity at room temperature can be obtained only for sample sizes of the order of 1 mm, something not considered previously. This unusual requirement is because collective phonon excitations, and not single phonons, are the main heat carriers in these materials; these excitations are characterized by mean free paths of the order of hundreds of micrometers. As a result, even Fourier's law becomes questionable in typical sample sizes, because its statistical nature makes it applicable only in the thermodynamic limit to systems larger than a few mean free paths. Finally, we discuss the effects of isotopic disorder, strain, and chemical functionalization on thermal performance. Only chemical functionalization is found to play an important role, decreasing the conductivity by a factor of 2 in hydrogenated graphene, and by 1 order of magnitude in fluorogenated graphene.

Length-dependent thermal conductivity in suspended single-layer graphene

Graphene exhibits extraordinary electronic and mechanical properties, and extremely high thermal conductivity. Being a very stable atomically thick membrane that can be suspended between two leads, graphene provides a perfect test platform for studying thermal conductivity in two-dimensional systems, which is of primary importance for phonon transport in low-dimensional materials. Here we report experimental measurements and non-equilibrium molecular dynamics simulations of thermal conduction in suspended single-layer graphene as a function of both temperature and sample length. Interestingly and in contrast to bulk materials, at 300 K, thermal conductivity keeps increasing and remains logarithmically divergent with sample length even for sample lengths much larger than the average phonon mean free path. This result is a consequence of the two-dimensional nature of phonons in graphene, and provides fundamental understanding of thermal transport in two-dimensional materials.

Anomalous heat diffusion

Consider anomalous energy spread in solid phases, i.e., <Δx(2)(t)>E≡∫(x-<x>E)(2)ρE(x,t)dx∝t(β), as induced by a small initial excess energy perturbation distribution ρE(x,t=0) away from equilibrium. The second derivative of this variance of the nonequilibrium excess energy distribution is shown to rigorously obey the intriguing relation d(2)<Δx(2)(t)>E/dt2=2CJJ(t)/(kBT(2)c), where CJJ(t) equals the thermal equilibrium total heat flux autocorrelation function and c is the specific volumetric heat capacity. Its integral assumes a time-local Helfand-like relation. Given that the averaged nonequilibrium heat flux is governed by an anomalous heat conductivity, the energy diffusion scaling determines a corresponding anomalous thermal conductivity scaling behavior.

Logarithmic divergent thermal conductivity in two-dimensional nonlinear lattices

Heat conduction in three two-dimensional (2D) momentum-conserving nonlinear lattices are numerically calculated via both nonequilibrium heat-bath and equilibrium Green-Kubo algorithms. It is expected by mainstream theories that heat conduction in such 2D lattices is divergent and the thermal conductivity κ increases with lattice length N logarithmically. Our simulations for the purely quartic lattice firmly confirm it. However, very robust finite-size effects are observed in the calculations for the other two lattices, which well explain some existing studies and imply the extreme difficulties in observing their true asymptotic behaviors with affordable computation resources.

Anomalous size dependence of the thermal conductivity of graphene ribbons

We investigated the thermal conductivity K of graphene ribbons and graphite slabs as the function of their lateral dimensions. Our theoretical model considered the anharmonic three-phonon processes to the second-order and included the angle-dependent phonon scattering from the ribbon edges. It was found that the long mean free path of the long-wavelength acoustic phonons in graphene can lead to an unusual nonmonotonic dependence of the thermal conductivity on the length L of a ribbon. The effect is pronounced for the ribbons with the smooth edges (specularity parameter p > 0.5). Our results also suggest that, contrary to what was previously thought, the bulk-like three-dimensional phonons in graphite make a rather substantial contribution to its in-plane thermal conductivity. The Umklapp-limited thermal conductivity of graphite slabs scales, for L below ∼30 μm, as log(L), while for larger L, the thermal conductivity approaches a finite value following the dependence K(0) - A × L(-1/2), where K(0) and A are parameters independent of the length. Our theoretical results clarify the scaling of the phonon thermal conductivity with the lateral sizes in graphene and graphite. The revealed anomalous dependence K(L) for the micrometer-size graphene ribbons can account for some of the discrepancy in reported experimental data for graphene.

Heat conduction in a three-dimensional momentum-conserving anharmonic lattice

Heat conduction in three-dimensional anharmonic lattices was numerically studied by the Green-Kubo theory. For a given lattice width W, a dimensional crossover is generally observed to occur at a W-dependent threshold of the lattice length. Lattices shorter than W will display a 3D behavior while lattices longer than W will display a 1D behavior. In the 3D regime, the heat current autocorrelation function was found to show a power-law decay as a function of the time lag τ as τ^{β} with β=-1.2. This indicates normal heat conduction. However, the decay exponent deviates significantly from the conventional theoretical value of β=-1.5. A flat power spectrum S(ω) of the global heat current in the low-frequency limit was also observed in the 3D regime. This provides not only an alternative verification of normal heat conduction but also a clear physical insight into its origin.

Heat conduction in two-dimensional disk models

We study the heat conduction problem in two-dimensional (2D) lattice models of disk shape consisting of two circular heat baths with radius r1 and r2 (r1<r2) , located concentrically at the center and the edge of the disk. Compared with the lattice models of rectangle shape adopted in previous studies, the main advantage of the disk models is that they have an unambiguous 2D dimensionality. The Fermi-Pasta-Ulam interaction of β type and the ϕ4 system are considered, respectively, as momentum conserving and nonconserving prototypes. In the former we find that in the range of the system size investigated, the heat conductivity κ depends on the system size L=r2-r1 as κ∼(ln L)α with α being a function of r1/r2. In particular, in the limit of r1/r2→1 we have α→1 , i.e., a logarithmic dependence of κ on L , which is in agreement with the prediction of existing theories. In the momentum nonconserving ϕ4 system the heat conductivity converges to a finite value as the system size is increased.

Dimensional crossover of thermal transport in few-layer graphene

Graphene, in addition to its unique electronic and optical properties, reveals unusually high thermal conductivity. The fact that the thermal conductivity of large enough graphene sheets should be higher than that of basal planes of bulk graphite was predicted theoretically by Klemens. However, the exact mechanisms behind the drastic alteration of a material's intrinsic ability to conduct heat as its dimensionality changes from two to three dimensions remain elusive. The recent availability of high-quality few-layer graphene (FLG) materials allowed us to study dimensional crossover experimentally. Here we show that the room-temperature thermal conductivity changes from approximately 2,800 to approximately 1,300 W m(-1) K(-1) as the number of atomic planes in FLG increases from 2 to 4. We explained the observed evolution from two dimensions to bulk by the cross-plane coupling of the low-energy phonons and changes in the phonon Umklapp scattering. The obtained results shed light on heat conduction in low-dimensional materials and may open up FLG applications in thermal management of nanoelectronics.

Breakdown of Fourier's law in nanotube thermal conductors

We present experimental evidence that the room temperature thermal conductivity (kappa) of individual multiwalled carbon and boron-nitride nanotubes does not obey Fourier's empirical law of thermal conduction. Because of isotopic disorder, kappa's of carbon nanotubes and boron-nitride nanotubes show different length dependence behavior. Moreover, for these systems we find that Fourier's law is violated even when the phonon mean free path is much shorter than the sample length.