The synthesis and measurements of nanomaterials have yielded significant advances in the past decades. In the area of thermal conduction, the nanomaterials exhibit anomalous behavior such as size-dependent thermal conductivity, thermal rectification, and ultra-high thermoelectric properties. The theoretical understanding and modeling on these behaviors are much desired. In this chapter, we study the thermal conduction in nanomaterials through the thermomass theory, which models the heat transfer from a fluid mechanics viewpoint. The control equations of the equivalent mass of the thermal energy are formulated following the continuum mechanics principles, which give the general heat conduction law. It incorporates nonlinear effects such as spatial acceleration and boundary resistance, which can overcome the drawbacks of the traditional Fourier’s law in nanoscale systems. By the thermomass theory, we successfully model the size-dependent effective thermal conductivity in nanosystems. Furthermore, the thermal rectification as well as the thermoelectric enhancement in nanosystems is also discussed with the present framework.
- thermomass theory
- thermal conductivity
- thermal rectification
The Fourier law proposed in 1822  is the fundamental of thermal conduction. It indicates that the heat flux passing through a material is proportional to the local gradient of temperature
The reduced thermal conductivity of nanofilms is a disadvantage for the heat dissipation in IC chips or semiconductor lasers. Nevertheless, it is an advantage for the thermoelectric devices. Experiments showed that the silicon nanowires have very high figure of merit (ZT) [10, 11]. The nanocomposites also demonstrate considerable ZT benefiting from the nano-sized superlattice or grains significantly scatter the phonons and reduce the effective thermal conductivity [12, 13]. Therefore, a lot of effort has been made to fabricate materials with ultra-low thermal conductivity through nanotechnology with the target at high ZT for the applications in advanced heating and cooling, waste heat recovery , as well as solar thermoelectric generators .
Due to the fast growth of energy-related nanomaterial synthesis and its transition from laboratory to industrial applications, modeling the thermal conducting behavior in nanosystems is in urgent need. Ideally, it should rise from a perspective of characterizing the fundamental physics and approach to simply structured theory which can be conveniently used by engineers. Nevertheless, this goal has not been satisfactorily achieved and current research is paving toward it. The gray model proposed by Majumdar is a pioneer work in this path. It predicts the effective thermal conductivity as 
The phonon hydrodynamics [26–31] is another pathway to model the nanoscale heat conduction. It originates from the solving of linearized Boltzmann equation. An additional term representing the second order spatial derivative of heat flux, ∇2
Upon the abovementioned progresses and their defects, the development of better models characterizing heat conducting in nanomaterials should base on capturing the essential feature of its physics. In recent years, the thermomass theory has been developed in our group, which proposes a mechanical analysis framework for heat transfer [32–35]. The generalized heat conduction governing equations are established based on such analysis. In the following sections, we will present the application of thermomass theory in nanomaterial heat conduction. The size dependency of thermal conductivity, thermal rectification, and thermoelectric effects will be addressed.
2. Thermomass theory
In history, the nature of heat was regarded as either a fluid, that is, caloric theory. The caloric theory regards heat as a weightless, self-repulsive fluid. In the eighteenth and the first half of nineteenth centuries, the caloric theory was the mainstream theory. It was extinct after the mid-nineteenth century and replaced by the dynamic theory that the nature of heat is the random motion of particles in a body. In twentieth century, Einstein’s relativity theory introduced the well-known mass-energy equivalence relation,
Consider the dielectric solids, the phonons are the main heat carriers. In this case, the internal energy per unit volume,
It should be reminded that the frequently used expression for thermal conductivity of phonon systems, Eq. (2), is from the analogy between gas and heat carriers. The scattering of phonons induces resistance on heat transport. Generally, the scattering accounted for thermal resistance is the R processes, including the Umklapp scattering, defect scattering, and boundary scattering. These scattering events eliminate the quasi-momentum of phonons. The MFP defined in Eq. (2) refers to the traveled distance of a phonon between succeeding R scatterings. However, in ideal gas systems, the collision among gas molecules does not perish the momentums. Therefore, the R processes of phonons are more resemble to the collision of gas molecules to residential barriers. It is the case when a gas flows through a porous medium. The collision frequency between gas molecules and material skeleton determines the resistance experienced by the gas flow. In the porous flow, the Darcy’s law describes the effective flow velocity is proportional to the pressure gradient
The pressure gradient can be regarded as the driving force of flow. From a viewpoint of force balance, the driving force is actually balanced by the friction force. Thereby Eq. (7) essentially depicts that the friction force is proportional to the flow velocity. It is a general case in laminar flow.
In analogy to the gas flow in porous medium, the velocity of thermomass is defined as
where g is the group velocity of phonons. For bulk material, the friction experienced by thermomass can be extracted from Eq. (7). When discussing the nanosystems, the boundary effect needs to be considered. The Darcy’s law for porous flow was extended to Darcy-Brinkman relation when the boundary effect is nonnegligible [37, 38]
In large systems, the boundary effect is negligible. Then, Eq. (13) reduces to the Darcy’s law with the first term much more important than the second term on the right hand side.
When the spatial gradient and changing rate of physical quantities are not significant, the first and second terms in Eq. (10) can be neglected. In this case, Eq. (10) exhibits the balance between driving force and friction force. The heat conduction is steady in such a nonequilibrium system. Combining Eqs. (13) and (10) leads to
For the simplest case, g and
Eq. (17) is a generalization of Fourier law when boundary effect needs to be considered. It predicts the reduction of effective thermal conductivity in nanosystems by the additional resistance term. When the system size is bigger, the spatial gradient of
3. Phonon Boltzmann derivation
where k is the phonon velocity in one Cartesian direction. The collision, such as the phonon-phonon scattering, reshapes the phonon distribution function. In phonon theory, the collisions can be sorted to R and N processes. The R processes break the phonon quasi-momentum, while the N processes conserve it. In this sense, the collision operator can be simply formulated as
If the friction force in Eq. (10) only has the first term, which is linear to the thermomass velocity, Eq. (22) is identical to Eq. (10) except the coefficient 15/16 in ahead of the second term on the left hand side. This difference is caused by the Doppler Effect during the drift motion of phonon gas. From this perspective, the phonon gas is slightly different from the real gas. The phonon energy varies due to the dispersion, causing the “eclipse” of the convection term. In a nondispersive medium, the frequency is independent of
It indicates that with the diffusive boundary, the N processes induce a deviation from
Eq. (25) can be regarded as the first order Chapman-Enskog expansion  of the phonon distribution function. In fluid mechanics, the viscous term in Navier-Stokes equation can be derived from the first order Chapman-Enskog expansion of the state distribution function of fluid molecular. Without the Chapman-Enskog expansion, the solution of Boltzmann equation gives the Euler equation, which is the dynamic equation without the viscous dissipation. This case happens when the interested region is far away from the boundary, or the boundary layer thickness is negligible compared with the flow region, like the large Reynolds number flow around the aircrafts. The difference between the thermomass flow and ordinary gas flow is that the R processes causes residential friction forces to the flow, which makes the transfer diffusive. In low temperature crystals, or low dimensional materials, such as graphene, the R processes can be rare. Then the heat conduction will exhibit obvious hydrodynamic behaviors. Therefore, based on the phonon Boltzmann derivation, the value of
4. Phonon gas flow in Si nanosystems
Based on Eq. (25) we can calculate the effective thermal conductivity of nanosystems. The silicon nanofilms and nanowires are investigated here because the experimental results are available for comparison. The geometries of nanofilms and nanowires are shown in Figure 1. The direction of heat conduction is in-plane for nanofilms and longitudinal for nanowires.
Eqs. (26) and (28) show the heat flux is nonuniform at the cross-section. If the system size is much larger than
The analytical derivation of Eqs. (26)–(30) is based on the assumption that the
In this way, the effective MFPs in nanosystems can be obtained by integrating over the sphere angle. For nanofilms, the local value of MFPs is 
Therefore, the MFPs are significantly shortened in nanosystems. It reveals that the boundary has dual effects on heat conduction in nanosystems. First, the second spatial derivative of heat flux, which represents the viscous effect of phonon gas, imposes additional resistance on heat transfer due to the nonslip boundary condition. Second, the collision on boundary changes the effective MFPs. This effect is similar to the rarefaction of gas flow in high Kn case. By accounting both the dual effects, the thermal conduction in nanosystems is described as
It is worth noting that in fluid mechanics, the rarefaction is not necessarily happened at the same time of viscous effect based on the Darcy-Brinkman relation. Consider the water flow in porous material. The permeability of porous flow is determined by the size of pores, which typically is in the order of micrometers. The MFP among water molecule is typically subnanometer. Therefore, the square root of permeability differs much from the MFP. The effects of Darcy-Brinkman boundary layer and rarefaction can be unconjugated. On the other hand, if the fluid is replaced by gas, the MFP of fluid could be comparable to the square root of permeability. In this case, the Darcy-Brinkman boundary layer and the rarefaction should be considered simultaneously. For the phonon gas flow, the relative magnitude of
The numerical solution of Eq. (34) gives the effective thermal conductivity for Si nanofilms and nanowires at room temperature, as shown in Figure 2. The physical properties are adopted as
5. Thermal rectification in nanosystems
Thermal rectification refers that the heat conduction in one direction of the device leads to higher heat flux than following the opposite direction, even though the same temperature difference is applied. It currently raises much interest since the first experimental report by carbon nanotubes . The thermal rectification effect is anticipated to realize thermal diode , thermal logic gate , or thermal transistors [46, 47]. Though much effort has been paid for searching useful mechanisms and realizing considerable rectification ratio, the ambitious goal that controlling heat as electricity is still far away .
The mechanism of thermal rectification has been widely studied. It is found that various effects can induce rectification, such as the different temperature dependences of the thermal conductivity at the different parts of the device , the asymmetric transmission rates of phonons across the interfaces , and the temperature dependence of electromagnetic resonances . Here, another rectification mechanism is proposed through the thermomass theory, following an analogy to fluid mechanics. In Navier-Stokes equations, the convective acceleration term indicates when the fluid experiences speed up or slow down. Therefore, if the cross-section area of a flow channel is changing (e.g. the trapezoidal channel), the flow rate under the same pressure difference is different in the convergent direction or in the divergent direction. In the convergent direction, the channel serves as a nozzle, which accelerates the fluid and converts part of its potential energy to the kinetic energy. In the divergent direction, the channel serves as a diffuser, which decelerates the fluid and converts part of its kinetic energy to the potential energy. The acceleration of fluid increases the velocity head and consumes the dynamic head of flow. Therefore, the total fluid flux in the convergent direction will be less than that in the divergent direction. In terms of thermal conduction, it means that with the same temperature difference between the heat source and sink, the total heat flux in the wide-to-narrow direction is smaller than that in the narrow-to-wide direction, which is the thermal rectification. Nevertheless, it should be stressed that for a flow channel with large angle of divergence, the flow separation could happen when the fluid velocity is high. In case of flow separation, the effective resistance of the diffuser will be much increased. It may cause the total heat flux in the wide-to-narrow direction larger than that in the narrow-to-wide direction, that is, the reverse of rectification.
In steady state, the generalized conduction law, Eq. (10), can be reformulated as
The difference between Eqs. (35) and (25) is the additional convective term, −
The sign of the first term of
To enhance the thermal rectification, the directional sensitive part in Eq. (36) should be amplified over the directional non-sensitive part. If the diffusive boundary condition is replaced with slip boundary condition, or the system size is large compared with the boundary layer, the Laplacian term of heat flux can be neglected. In room temperature, the second term of
Consider a silicon ribbon with the average temperature 300 K. Assume that
6. Thermoelectricity of nanosystems
The ZT for nanomaterials could be much enhanced [10–13]. The mechanism of such enhancement can be that the nanostructures reduce the thermal conductivity by strong phonon-boundary scattering while maintaining the electrical conductivity. Although a lot of work has been done in searching high ZT materials through nanotechnology, the thermodynamic analysis and the role of nonlocal and nonlinear transports, which are highly possible to happen in nanosystems, are not fully discussed [54, 55]. In recent years, the nonlocal effects raised by the MFP reduction due to geometry constraint , the electron and phonon temperature , and the breakdown of Onsager reciprocal relation (ORR) [58, 59] in nanosystems have been investigated from the framework of extended irreversible thermodynamics (EIT). These works showed that the nonlinear and nonlocal effects influence the efficiency of devices. The breakdown of ORR not only possesses theoretical importance but also shed light on approaches to further increase efficiency.
Here, we analyze the thermoelectric effect from the thermomass theory perspective. There could be various effects when the individual motion of phonon gas and electron gas is separately considered. The most apparent one is the energy exchange between phonons and electrons . In a one-dimensional thermoelectric medium, the conservation of energy gives
The second term on the left hand side is nonzero because of the energy conversion. It increases the spatial inertia of thermomass. For simplicity, we do not consider the Brinkman extension of the friction force and assume the material cross-section is constant, and then Eq. (40) turns to
Compared with Eq. (37), the first term of
Since ZT is
In this chapter, we present a mechanical analysis on the thermal conduction in nanosystems with the thermomass theory. Firstly, the boundary resistance in nanosystems on heat flow is modeled with the Darcy-Brinkman analogy. The permeability of thermomass in materials is derived based on the phonon Boltzmann equation. The size-dependent effective thermal conductivity of Si nanosystems thereby is accurately predicted with the present model. Then, the spatial inertia effect of thermomass is shown to induce the thermal rectification in asymmetry nanosystems. The predicted rectification ratio can be as high as 32.3% in a trapezoidal Si nanoribbon. Finally, the energy conversion in thermoelectric devices can be coupled with the spatial inertia of thermomass flow. The ZT tends to be increased in case of a thermoelectric generator.
This work was financially supported by the National Natural Science Foundation of China (nos. 51136001, 51356001) and the Tsinghua University Initiative Scientific Research Program.