Revisit nonequilibrium thermodynamics based on thermomass theory and its applications in nanosystems

1.1 Local equilibrium thermodynamics

Equilibrium thermodynamics, developed in the 19th century, is concerned with the macroscopic properties of matter at near-equilibrium, but it only involves the initial and final equilibrium states and cannot describe the process itself. In 1931, Onsager [1] used fluctuation linear regression and the microscopic reversibility assumption to derive general reciprocity relationships that apply to transport processes like heat conduction, electrical conduction, and diffusion. In 1961, Prigogine [2], [3] discovered a dissipative system that operates in an environment with which it exchanges energy and matter, often far from thermodynamic equilibrium. These two pioneering works, along with a series of other developments [4], gave rise to local equilibrium thermodynamics, also known as classical irreversible thermodynamics (CIT).

In CIT, the entropy production obtains

which has a structure of a bilinear form in the generalized flux, J, and the generalized force, X. For general irreversible energy transfer processes, Eq. (1) can be written as

where q is the heat flux, P ^v is the stress tensor, u _f is the fluid velocity, J _k is the mass component diffusion flux, μ _k is the chemical potential, and i is the electric potential. Each term represents the entropy production induced by heat conduction, momentum diffusion, mass component diffusion, and electrical conduction, respectively.

1.2 Extended irreversible thermodynamics (EIT)

The local equilibrium hypothesis implies large time and space scales, so high-frequency processes and non-Fourier heat conduction cannot be adequately described by CIT, such as nanomaterials, neutron scattering, and superfluids [5]. Therefore, the proposal of extended irreversible thermodynamics (EIT) goes beyond the local equilibrium assumption and introduces dissipative fluxes (heat flux, diffusion flux, electric flux, and viscous pressure) as independent variables to describe these phenomena [6]. The key to constructing EIT is the establishment of flux evolution equation and the reconstruction of non-equilibrium entropy.

Inserting the CV model [7]–[9],

where τ is the relaxation time and κ is the thermal conductivity, into the classical expression of the entropy production, Eq. (2), it can be found that

which no longer remains positive definite due to the presence of linear terms in the equation. When Fourier’s law holds, the entropy not only depends on the temperature, T, also depends on the heat flux, q. To eliminate this paradox, EIT considers the heat flux, q, as an independent variable. Combined the CV model the extended entropy production is

To obtain an evolution equation for q compatible with the positive value of σ ^S by assuming that the force X is linearly related to q, i.e. the corresponding positive extended entropy production is

By altering entropy and entropy production as illustrated in Figure 1, EIT eliminates the paradox of negative entropy production in non-Fourier heat conduction. Therefore, it is necessary to redefine the thermodynamics temperature so that the nonequilibrium temperature, θ, in EIT is

where T _eq is the temperature at equilibrium state, q is the local heat flux, κ is the thermal conductivity, C is the specific heat capacity, ρ is the density of material, and τ is the relaxation time. Eq. (7) indicates that θ is lower than T _eq . However, EIT does not give an explicit physical definition of generalized forces and fluxes. The nonequilibrium temperature, θ, is based on a mathematical derivation of the extended entropy production and requires more physical discussions. Thus, more fundamental research on nonequilibrium thermodynamics is required to address non-Fourier heat conduction.

Figure 1:

Entropy evolutions in an isolated 1D system of the extended entropy (solid line) and the conventional equilibrium entropy (dash line) [10].

In nanosystems, the limited velocity of energy carriers (phonons) is responsible for the limiting heat propagation speed. The time required for energy carriers to accelerate is disregarded by Fourier’s law. For example, Tzou [11] experimentally observed the lagging behavior of heat transfer in small-scale and fast-transient conditions, which further proves that heat has inertia [1], [12]. Therefore, Fourier’s law fails because the phonon transport is only partially ballistic in certain temporal and geographical scales. Some transport equations were derived macroscopically by EIT and applied to study the heat conduction in nanowires [13]–[18]. Meanwhile, several simulations and theoretical studies revealed the one-dimensional nanosystem’s anomalous length dependence of thermal conductivity [19]–[26]. Although low-dimensional materials have longer phonon mean free paths than conventional materials, which causes their thermal conductivity to increase with length over a larger size range. However, Fourier’s law is still the foundation for these computations and measurements of thermal conductivity. This method of approximating thermal conductivity may not be appropriate when Fourier’s law fails. Hence, it is highly desirable to have a more comprehensive heat conduction model that is appropriate for non-Fourier heat conduction and has explicit macroscopic physical meaning.

1.3 Thermomass theory

Before the 20th century, it was generally accepted that the heat transfer process was simply an energy transfer without the transport of substance [27]. In contemporary thermodynamics, via Einstein’s special relativity, Guo et al. [28]–[30] determined the equivalent mass (also known as the thermomass) of the thermal energy as a component of the system’s invariant mass. The inertia effect of thermomass is attributed to the nonlocal and nonlinear heat conduction. The variation of effective thermal conductivity with system size and temperature at steady state is investigated using the governing generalized heat conduction equation based on the thermomass gas model [31], [32], which is consistent with the experimental results [33]. Recently, Nie et al. [34] clarified the mathematical rationality of generalized heat conduction equation in the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework and revealed that thermomass theory gives the physical significance of Hamiltonian energy and dissipation potential in heat conduction under GENERIC framework. The mass balance equation of a thermomass gas:

The momentum balance equation of a thermomass gas:

with

where ρ _T is the density of the thermomass gas, u _T is the drift velocity of the thermomass gas, P _T is the thermomass pressure, and f _T is the resistance force per unit volume of the thermomass gas flowing through the medium. c is the velocity of light and α is a proportional parameter depending on the state of medium. Eq. (8) and (9) describe the motion of the thermomass gas without any other artificial hypothesis. Refer to the detailed derivation process [31], [32], the generalized heat conduction equation can be written as

where

τ _T is a characteristic time between the temperature gradient and heat flux. l is a characteristic length measuring the spatial inertia of thermomass. b is a dimensionless parameter for characterizing flow compressibility of thermomass gas. Eq. (11) shows the transient inertia effect in the 1st term. The 2nd and 3rd terms represent the spatial inertia effects. The driving force is represented by the 4th term, and the resistance force by the final term. The Fourier’s law is recovered by Eq. (11) if all the effects of inertia are disregarded. The CV model is what remains when the spatial inertia effect alone is negligible.

In this review, we first briefly introduce the extended irreversible thermodynamics (EIT) and the generalized heat conduction model based on thermomass theory. Then, we reshape the concepts such as entropy production, temperature and Onsager reciprocal relation (ORR) based on EIT and thermomass theory. Finally, we review recent applications of thermomass theory for thermal conductivity, thermoelectric effect, and thermal rectification effect in nanosystems.

3.1 The zeroth law

The zeroth law gives the definition of thermal equilibrium and allows the temperature to become the measurable quantity. According to Fourier’s law, heat flux can only occur between two systems with different temperatures, T _eq . EIT proposes the nonequilibrium temperature, θ, as the criterion of thermal equilibrium [5], [6]. According to Eq. (7), θ will reduce due to the presence of heat flux, q, which means that heat flux can occur among systems with the same T _eq . based on the thermomass model, the heat flux is driven by the static pressure gradient of thermomass gas, ∇p _T . Thus, we can obtain a relation between the total temperature of thermomass gas, T _t and the static temperature, T,

Equation (18) shows that the static temperature, T, is smaller than the total temperature of thermomass gas, T _t , which means that the heat conduction occurs as long as there is a difference of static temperature between two systems. Analogous to EIT, the static temperature, T, corresponds to the nonequilibrium temperature, θ, while the total temperature of thermomass gas, T _t , corresponds to the equilibrium temperature, T _eq , which carries all the energy of system. When the heat flux is stagnant, θ is equal to T _eq and T is equal to T _t .

3.2 The second law

In Section 2, thermomass theory reshapes the expression of entropy production by the resistance force and the drift velocity. Hence, we can also reshape the temperature by using the total temperature of thermomass gas, T _t , to measure the internal energy of system. From the classical relation among temperature, entropy, and internal energy, T ⁻¹ = ds/de, we can obtain

We can find that Eq. (19) is similar to the EIT’s derivation, Eq. (7). However, there is a small distinction between the relationship between the static temperature, T, and the stagnant temperature, T _t , and that between the nonequilibrium temperature, θ, and the equilibrium temperature, T _eq . There is no difference between entropy of the nonequilibrium and that of static temperatures. It is the internal energy expression that differs. In EIT, the definition of internal energy is based on the equilibrium temperature, T _eq , which includes all molecular energies of phonon gas. But in thermomass theory, the internal energy is defined as the static temperature, T without the drift kinetic energy. Therefore, the temperature produced by both the thermomass theory and the EIT is lower than the total or equilibrium temperature; the reason for this discrepancy is the way internal energy is described differently by the two theories.

5.1 Thermal conductivity of nanomaterials

As mentioned above, when the thermal inertia effect cannot be ignored, the generalized heat conduction equation can be used to study non-Fourier heat conduction, such as thermal conductivity of nanomaterials, size dependence of thermal conductivity, and thermal rectification.

Cao et al. [28] investigated the thermal conductivity of a single carbon nanotube (CNT) with different lengths based on the thermomass gas model. They used a CNT with an intrinsic thermal conductivity of 5000 W/mK as an example, and the temperature difference between the two ends was assumed to be 20 K. They predicted the apparent thermal conductivity of CNT by

where κ _app is the apparent thermal conductivity and κ is the intrinsic thermal conductivity. The difference between the apparent and intrinsic thermal conductivities is due to the spatial inertia of thermomass.

Figure 2 indicates that the apparent thermal conductivity of CNT increases with increasing length. Eq. (28) reveals that the apparent thermal conductivity is always smaller than intrinsic thermal conductivity of materials. The length dependence of thermal conductivity of CNT agrees with the MD simulations [22], [40], which shows an exponentially increasing trend. Recent several theoretical studies [19], [20], [24] and experimental observations [25], [26] reveal the anomalous heat conduction in one-dimensional system that there is a presence of divergent thermal conductivity with length. The thermomass model could be used as a mesoscopic theory to explore the physical mechanism of anomalous heat conduction from micro to macro.

Figure 2:

The predicted apparent thermal conductivity of CNT with incremental length. Reprinted with permissions from AIP Publishing [28].

At nanoscales, the thermal conductivity of materials is limited by phonon mean free paths (MFP). In this case, the rarefication effect of phonon gas should be considered to calculate the effective thermal conductivity. To address the rarefication effect of phonon gas, Dong et al. [41] proposed a macroscopic heat conduction based on the phonon gas model, to which a second order resistance term is added in the generalized heat conduction equation. They were able to derive an explicit expression for effective thermal conductivity of Si nanomaterials.

For a nanofilm, the effective thermal conductivity is

For a nanowire, the effective thermal conductivity is

where λ _E is the characteristic length depending on the material properties and ambient temperature. Figure (3) demonstrates that the effective thermal conductivities of Si nanofilm and nanowire increase with length and approach the bulk thermal conductivities. The size dependence of thermal conductivity predicted by the generalized heat conduction model agrees with the experimental results, where the additional second order resistance term is applied to assess the rarefication effect of phonon gas caused by the enhanced viscous friction on nanoscale boundaries.

Figure 3:

The length dependence of the effective thermal conductivity (κ _eff ) of Si nanofilm [dash line, Eq. (29)] and nanowire [solid line, Eq. (30)] predicted by the modified generalized heat conduction model. The solid circles and triangles are the experimental results of nanofilm [42], [43] and nanowire [44], respectively. Reprinted with permissions from Elsevier [41].

5.2 Thermoelectric effect of nanomaterials

The development of nanotechnology enhances the thermoelectric figure of merit (ZT) of nanomaterials [45]–[48]. This enhanced mechanism may be due to the fact that while the nanostructure maintains the electric conductivity, the phonon boundary scattering also reduces the thermal conductivity of the material. The thermomass theory gives a new perspective to analyze the thermoelectric effect. Energy exchange occurs between electrons and phonons [49],

where I is the electric current and E is the electric field. Combining the generalized heat conduction model without the inertia effect and the dimensionless figure of merit, ZT = S ² ρ ⁻¹ κ ⁻¹ T. When IE > 0, electrical energy is converted to thermal energy, and the system is a typical thermoelectric cooler. The heat flux is also hindered, and the thermoelectric ZT decreases. When IE < 0, thermal energy is converted into electrical energy, which is a thermoelectric generator. The heat flux is further increased, and the effective thermoelectric ZT increases. In addition, the inertial effect may favor the ZT of the device.

Rogolino et al. [50], proposed a nonlinear equation of thermoelectric coupling based on thermomass theory and used it to evaluate the efficiency of the thermoelectric generator and investigate the effect of current on thermoelectric efficiency of device.

Cimmelli and Rogolino [51] examined the efficiency of a thermoelectric energy generator composed of a silicon-germanium alloy using the generalized heat conduction model and talked about nonlocal and nonlinear heat transport in nanosystems. They looked into the dependence of the thermoelectric efficiency on composition and temperature differential in addition to experimentally determining the thermal conductivity of a nanowire as a function of temperature and composition.

Youssef and AI-Lehaibi [52] drew on thermomass theory to limit the propagated speed of thermal and mechanical waves to set up an extended thermoelasticity model. It brings thermoelastic materials closer to their true physical behavior by weakening the coupling relationship between mechanical waves and thermal energy.

5.3 Thermal rectification in nanosystems

Recently thermal rectification effect in nanosystems has attracted wide interesting. Researchers have constructed a variety of thermal rectification devices through molecular dynamics simulation and experiments [53]–[55], among which the most important forms are asymmetric nanostructures. For example, mass graded carbon nanotube [56], carbon nanocone [57], asymmetric graphene ribbon [58], [59], low-dimensional heterojunction [60], asymmetric molecular bridge [61] and so on. The mechanism of thermal rectification phenomenon is attributed to the asymmetry of phonon frequency-energy spectrum coupling [62], but there is no definite quantitative studies. A potential mechanism of thermal rectification is proposed by combining thermomass theory and hydrodynamics. The convective acceleration term in the Navier–Stokes equation represents when the fluid accelerates or decelerates. Therefore, when the cross-sectional area of the flow channel is changed, the flow at the same pressure difference is different in the anteroposterior direction. For a medium with uniformly varying cross section, the equivalent thermal resistance, ϒ, can be expressed as [49]

where the first term on the right side of Eq. (32) is the inertial effect of thermomass. When the heat flow direction is opposite, this part will change positively or negatively. The second term on the right side of Eq. (32) is resistance force of thermomass, which is mainly determined by the shape of medium but not by the direction of heat flux. When the heat flux is from the narrow to wide (NTW), A ₂ > A ₁ , inertial thermal resistance is less than 0; When the heat flux is from the wide to narrow (WTN), A ₁ > A ₂ , inertial thermal resistance is greater than 0. Figure 4(a) shows a trapezoidal silicon nanofilm with thickness D of 50 nm, width of narrow end, L _N = 50 nm, width of narrow end, L _W = 300 nm, and length of nanofilm, L = 500 nm. The thermal rectification ratio, η, and wall angle, θ, are defined by

Figure 4:

Schematic of thermal rectification model based on thermomass theory: (a) Thermal rectification in the trapezoidal nanoribbon. (b) Thermal rectification ratio changes with wall angle and temperature.

Figure 4(b) shows that the rectification ratio decreases with the increasing temperature. The larger the wall angle, the larger the rectification ratio. When the wall angle is 18.9°, the rectification ratio can reach nearly 40 % at 220 K. However, when the wall angle is large, the flow separation phenomenon at the wall may occur, and the rectification ratio may sharply reduce or even become negative.

When the thermal inertia force is considered, the total equivalent thermal resistance of an asymmetric conductor contains inertial thermal resistance that changes with the direction of heat flow, resulting in thermal rectification. This is a steady-state non-Fourier heat conduction phenomenon caused by the inertia force of the thermomass. The equivalent thermal resistance in NTW mode is smaller than that in WTN mode, which is consistent with the experimental results in silicon nanofilms [63]. The rectification effect is more significant at lower temperatures and increases with the increase of wall angle. If flow separation occurs, the equivalent thermal resistance in NTW mode may be higher than that in WTN mode. Flow separation is more likely to occur when the wall angle and the heat flow are large, which is consistent with the results of MD simulation [58]. However, in the real and complex heterogeneous nanosystems, more factors will affect the rectification effect, such as phonon MFP, phonon boundary scattering, ballistic-diffusive phonon coupling, and negative differential thermal resistance, etc.