1. Introduction
Nanotechnology has changed our vision, expectations and abilities to control all of the material world. It offers a huge amount of potential applications and solutions that in the recent past were only possible in the realm of scientific fiction. Its usefulness has been known for many years, especially in microelectronics and, more recently, in manufacturing of nano-electromechanical-systems (NEMS): systems with less space, material, energy consumption and usually higher performance can be actually realized. In particular, current research in nanoscaled electronic devices focuses the attention on carbon nanotubes, quantum wires and quantum dots, where the additional confinement of electrons is used to make compact high-performance components.
From the theoretical point of view, the use of nanotechnologies requires revising some well-known theories, as for example the classical Fourier law. Experimental evidence, in fact, clearly shows that it is completely inefficient to describe accurately heat transport at the nanometer length scale [
1,
2,
3,
4,
5,
6,
7]. Several theories have been developed to describe heat transport in nanostructured materials [
8,
9,
10,
11,
12,
13,
14,
15].
In order to analyze thermoelectric effects, in [
16] the authors developed a hydrodynamic model accounting for the possibility that both electrons and phonons may contribute to the heat transport. In particular, they assumed that the local heat flux
q just arises from two contributions: the partial heat flux
due to the phonons and the partial heat flux
due to the electrons, in such a way that:
The aforementioned model can be applied to describe situations in which both a voltage at the ends of a conductor is created by a temperature difference (Seebeck effect) and a heat flux is generated by an electric current (Peltier effect). However, for common materials, these direct conversions are not always efficient, since they only show limited thermoelectric properties,
i.e., very small values of the Seebeck and Peltier material parameters. In such a case, for example, the Seebeck effect is only able to generate an electric current that is negligible, and in the absence of nonlinear effects, the evolution of the partial heat fluxes
and
in Equation (
1) is ruled by the following equations:
wherein
and
are the relaxation time, the mean-free path and the thermal conductivity of phonons and electrons, respectively, and
T is the absolute temperature. These equations account for nonlocal effects through the mean-free paths
and
, and for memory effects through the relaxation times
and
. They are still far from the standard form of the equations of linear heat-conduction theory with nonlocal and memory effects [
17], since they contain information on the partial heat fluxes, relaxation times, thermal conductivities and mean-free paths.
The system above must be coupled to the local balance of energy, which, in the absence of heat sources and with negligible electric current, reads:
with
the specific heat of the conductor.
Although more general situations, allowing for different temperatures for phonons and electrons, could be considered [
18,
19], in the present paper, we limit ourselves to the case in which the equations above hold. As we will see later, this assumption allows us to conserve a sufficient generality and reduces the calculations to a simpler level. Under the hypotheses above, here we study the radial heat transfer from a hot source surrounded by a circular layer and point out some non-standard characteristics of the temperature behavior.
The paper displays the following layout.
In
Section 2, we evaluate the system of Equations (
2) and (
3) in steady states, in order to obtain the differential equation for the temperature profile. In the special case of a two-dimensional system with a radial symmetry, that equation is solved numerically, and the influence of nonlocal effects on the solution is pointed out.
In
Section 3, we investigate the thermodynamic compatibility of the results obtained in
Section 2 by calculating numerically the local entropy production as a function of the distance from the heat source. We show that such a quantity is positive everywhere in the system, according to the prediction of the second law of thermodynamics.
In
Section 4, we draw the main conclusions and discuss possible developments of the present investigation.
3. Thermodynamic Compatibility
The behavior shown in
Figure 3 is anomalous, since the temperature profile in the layer is not everywhere decreasing with the radial distance, as it would be expected from the classical Fourier law [
23]. Since in the present problem, the only heat source is the hot device at
, that behavior (arising from the model Equation (
7)) points out that in the circular layer, there are regions wherein the heat is flowing from colder points to hotter ones.
In order to check the physical admissibility of the results plotted in
Figure 3, in the present section, we analyze them in view of the second law of thermodynamics. Therefore, let us consider the local balance of the entropy
with
s as the entropy per unit volume,
as the entropy flux in the absence of the electric-current density [
19] and
as the entropy production per unit volume.
The second law of thermodynamics states that
has to be always non-negative along arbitrary thermodynamic processes [
24]. Since in steady states from Equation (
18), it follows
, then the results of
Figure 3 are admissible on physical grounds if, and only if, the divergence of the entropy flux is non-negative everywhere in the layer. Indeed, by using the previous results for
,
and
, the occurrence of that condition may be easily checked once the following nondimensional version of the constitutive relation in Equation (
19) is introduced:
with
.
Figure 4 plots the behavior of the nondimensional entropy production (
i.e., the divergence of
) as a function of the radial distance from the hot device.
Figure 4.
Behavior of
versus the radial distance
in the circular layer that surrounds the hot device: theoretical results arising from the spatial derivatives of Equation (
20). The
y-axis in the figure is in a logarithmic scale.
Figure 4.
Behavior of
versus the radial distance
in the circular layer that surrounds the hot device: theoretical results arising from the spatial derivatives of Equation (
20). The
y-axis in the figure is in a logarithmic scale.
As it can be seen from
Figure 4, in the surrounding layer,
is everywhere positive. This proves that the hump in the temperature profile observed in
Figure 3 is physically admissible, since it agrees with the second law of thermodynamics.
In
Figure 4, it is also possible to see that
qualitatively behaves as
for increasing
, attaining its maximum and minimum at the same radial distances at which
also reaches its maximum and minimum values, respectively. Moreover,
is increasing for
. We feel that this result is due to the principal role played by the phonon contribution to the local heat flux in the particular problem analyzed here, since it is logical to suppose that
is mainly influenced by
. However, the behaviors observed in
Figure 2 and
Figure 4 deserve deeper investigations, especially with respect to their possible connection to the boundary conditions.
4. Conclusions
In the present paper, we have applied the nonlocal model Equations (
2) to describe heat transport in quasicrystalline materials [
25], namely, in materials that should be properly located at the border line between metals and semiconductors [
26]. For these materials, the heat transport is due to the combined flows of phonons and electrons [
27], which may also have different temperatures [
18,
19]. Assuming that both the phonon temperature and the electron one coincide, in the case of steady-state radial heat transfer from an inner hot device surrounded by an external circular layer, we have analyzed the consequences on the temperature profile of the contribution of nonlocal terms entering the temperature equation. Due to the special role played by nonlocal effects at nanoscale, we observed that the temperature profile shows an anomalous hump in the neighborhood of the hot device. The presence of that hump (see
Figure 3) suggests that there are some areas wherein the heat flows from colder points to the hotter ones. This result completely disregards that arising from the classical Fourier law [
23], which predicts that heat can only flow from the hotter zones to the colder ones. However, to check whether the theoretical predictions arising from Equation (
2) are physically admissible, or not, we have analyzed the temperature hump in view of the second law of thermodynamics.
Figure 4 shows that the entropy production is everywhere positive and points out that the temperature hump is only an apparent violation of the basic principles of thermodynamics.
There are some further problems related to the present investigation, which deserve consideration and that will be analyzed in our future research.
As we noticed in the
Section 1, it is also possible to consider more complex situations, with each of the heat carriers (phonons and electrons) endowed with its own temperature. Accounting for two different temperatures may be important, for instance, when a laser pulse hits the surface of a system. In such a case, initially, the electrons capture the main amount of the incoming energy with respect to the phonons. Subsequently, through electron-phonon collisions, they give a part of it to the phonons. As a further example, we mention those situations in which the electron mean-free path corresponding to electron-phonon collisions is long, so that one may have the so-called “hot electrons”, namely, a population of electrons whose average kinetic energy (
i.e., the kinetic temperature) is considerably higher than that of the phonons.
In the presence of two temperatures, the evolution of the partial heat fluxes
and
is governed by the following equations:
wherein
and
, respectively, mean the phonon and the electron contribution to the average temperature
T of the system [
28]. Both
and
may be related to the internal energy per unit volume of phonons
and to the internal energy per unit volume of electrons
, respectively, by means of the constitutive equations [
28]:
with
being the specific heat at constant volume due to the phonons, and
being the specific heat at constant volume due to the electrons, whereas the specific heat at constant volume of the whole system
is given by
.
From the practical point of view, Equations (
21) and (
22) may lead to some perplexities, since one may naturally wonder whether
and
are measurable quantities, or not [
29,
30]. Here we notice that in [
28] two possible strategies to measure those temperatures have been proposed.
Once Equations (
22) have been assumed and both the specific heats are supposed to be constant, then the following partial energy balances can be postulated:
For temperature-dependent specific heats, Equations (
23) become nonlinear. However, such a nonlinearity, which is due to the coefficients of the time derivatives of the partial temperatures, does not influence the temperature profile in steady states. Recalling that whenever both electrons and phonons contribute to the heat flow, the internal energy per unit volume
u of the whole system is given by
, then the use of Equation (
22) allows us to relate
and
to the average temperature
T of the system as:
once the thermodynamic relation
is used. Note further that the summation of Equation (
23) produces the local energy balance (
3).
The combination of Equations (
21) with Equations (
23) allows us, in principle, to derive the temperatures behaviors analytically, which may be interesting in practical applications. However, the solution of that problem is much more complex with respect to the case considered here. In [
19], the authors analyzed the consequences of accounting for Equations (
21) and (
23) in steady-state radial heat transfer from a point source when the partial contributions to the local heat flux are such that:
In this special case, both
and
display anomalous behaviors, which, however, are in accordance with the second law of thermodynamics [
19].
Relaxing the constitutive assumptions (
25), it would be interesting to obtain a numerical solution for both the temperature profiles
and
, in order to infer the different role played by phonons and electrons in quasicrystalline materials.
Another interesting problem is related to the efficiency of the thermoelectric coupling to two temperatures. In the presence of thermoelectric effects, the system of Equations (21)–(23) needs to be completed by suitable balance equations for the electric charge and for the electric current (see [
18,
19] for more details). Such a problem, when the nonlocal effects may be neglected, has been analyzed in [
28] with a uniform chemical potential due to the electric charge density and in [
31] with a nonuniform chemical potential due to the electric charge density. From those references, it follows that for practical applications, in order to enhance the efficiency of the thermoelectric energy conversion, it would be useful to have:
- (1)
the charge distribution as homogeneous as possible;
- (2)
the ratio as high as possible.
In our future research, we aim to extend the previous investigation to the case in which also nonlocal effects are taken into account, in order to check how these effects may influence the efficiency of thermoelectric devices.