Maximum power and the corresponding efficiency for a Carnot-like thermoelectric cycle based on fluctuation theorem.

Here, we investigate the maximum power and efficiency of thermoelectric generators through devising a set of protocols for the isothermal and adiabatic processes of thermoelectricity to build a Carnot-like thermoelectric cycle, with the analysis based on fluctuation theorem. The Carnot efficiency can be readily obtained for the quasistatic thermoelectric cycle with vanishing power. The maximum power-efficiency pair of the finite-time thermoelectric cycle is derived, which is found to have the identical form to that of Brownian motors characterized by the stochastic thermodynamics. However, it is of significant discrepancy compared to the linear-irreversible and endoreversible-thermodynamics based formulations. The distinction with the linear-irreversible-thermodynamics case could result from the difference in the definitions of Peltier and Seebeck coefficients in the thermoelectric cycle. As for the endoreversible thermodynamics, we argue the applicability of endoreversibility could be questionable for analyzing the Carnot-like thermoelectric cycle, due to the incompatibility of the endoreversible hypothesis that attributes the irreversibility to finite heat transfer with thermal reservoirs, though the distinction in the mathematical expressions can vanish with the assumption that the ratio of thermoelectric power factors at the high and low temperatures (γ) is equal to the square root of the temperature ratio, γ=sqrt[T_{L}/T_{H}] (this condition could significantly deviate from the practical case). Last, utilizing our models as a concise tool to evaluate the maximum power-efficiency pairs of realistic thermoelectric material, we present a case study on the n-type silicon.

Thermodynamic Derivation of the Reciprocal Relation of Thermoelectricity.

The Kelvin relation, relating the Seebeck coefficient and the Peltier coefficient, is a theoretical basis of thermoelectricity. It was first derived by Kelvin using a quasi-thermodynamic approach. However, Kelvin's approach was subjected to much criticism due to the rude neglect of irreversible factors. It was only later that a seemingly plausible proof of the Kelvin relation was given using the Onsager reciprocal relation with full consideration of irreversibility. Despite this, a critical issue remains. It is believed that the Seebeck and Peltier effects are thermodynamically reversible, and therefore, the Kelvin relation should also be independent of irreversibility. Kelvin's quasi-thermodynamic approach, although seemingly irrational, may well have touched on the essence of thermoelectricity. To avoid Kelvin's dilemma, this study conceives the physical scenarios of equilibrium thermodynamics to explore thermoelectricity. Unlike Kelvin's quasi-thermodynamic approach, here, a completely reversible thermodynamic approach is used to establish the reciprocal relations of thermoelectricity, on the basis of which the Kelvin relation is once again derived. Moreover, a direct thermodynamic derivation of the Onsager reciprocal relations for fluxes defined as the time derivative of an extensive state variable is given using the method of equilibrium thermodynamics. The present theory can be extended to other coupled phenomena.

Correspondence of the Symmetry of Thermodynamic Properties of Matter with the Symmetry of Equations of State.

Thermodynamics contains rich symmetries. These symmetries are usually considered independent of the structure of matter or the thermodynamic state where matter is located and, thus, highly universal. As Callen stated, the connection between the symmetry of fundamental laws and the macroscopic properties of matter is not trivially evident. However, this view is now being challenged. Recently, with symmetry to the ideal gas equation of state (EOS), an ideal dense matter EOS has been proposed, which has been verified to be in good agreement with the thermodynamic properties of high-density substances. This indicates that there is a certain symmetry between the thermodynamic properties of substances in their high- and low-density limits. This paper focuses on the distinctive features and the significance of this symmetry. It is a new class of symmetry that is dependent on the thermodynamic state of matter and can be incorporated into the existing symmetrical theoretical system of thermodynamics. A potential path for developing the EOS theory arising from this symmetry is discussed. EOS at high densities could be developed by correcting or extrapolating the ideal dense matter EOS based on this symmetry, which might fundamentally solve the difficulty of constructing EOS at high densities.

Thermoelectric Cycle and the Second Law of Thermodynamics.

In 2019, Schilling et al. claimed that they achieved the supercooling of a body without external intervention in their thermoelectric experiments, thus arguing that the second law of thermodynamics was bent. Kostic suggested that their claim lacked full comprehension of the second law of thermodynamics. A review of history shows that what Clausius referred to as the second law of thermodynamics is the theorem of the equivalence of transformations (unfairly ignored historically) in a reversible heat-work cycle, rather than "heat can never pass from a cold to a hot body without some other change" that was only viewed by Clausius as a natural phenomenon. Here, we propose the theorem of the equivalence of transformations for reversible thermoelectric cycles. The analysis shows that the supercooling phenomenon Schilling et al. observed is achieved by a reversible combined power-refrigeration cycle. According to the theorem of equivalence of transformations in reversible thermoelectric cycles, the reduction in body temperature to below the ambient temperature requires the body itself to have a higher initial temperature than ambient as compensation. Not only does the supercooling phenomenon not bend the second law, but it provides strong evidence of the second law.

A global equation-of-state model from mathematical interpolation between low- and high-density limits.

The ideal gas equation of state (EOS) model is a well-known low-density limiting model. Recently, an ideal dense matter EOS model for the high-density limit symmetric to the ideal gas model has been developed. Here, by mathematically interpolating between the ideal gas and ideal dense matter limiting models, we establish a global model containing two EOS in the form of P-V-T and P-S-T for arbitrary ranges of densities. Different from empirical or semi-empirical EOS, the coefficients in the global EOS have a clear physical meaning and can be determined from a priori knowledge. The proposed global model is thermodynamically consistent and continuous. It reduces to the ideal gas model when approaching the low-density limit and to the ideal dense matter model when approaching the high-density limit. Verifications for 4He show that the global model reproduces the large-range behavior of matter well, along with providing important insight into the nature of the large-range behavior. Compared to the third-order virial EOS and the Benedict-Webb-Rubin EOS, the global P-V-T EOS has higher descriptive accuracy with fewer coefficients over a wide range of data for N2. The global model is shown to work well in extreme applied sciences. It predicts a linear, inverse relationship between entropy and volume when the temperature-to-pressure ratio is constant, which can explain the entropy-production behavior in shock-Hugoniots.

Reversible reciprocal relation of thermoelectricity.

The first Kelvin relation, which states the Peltier coefficient should be equal to the product of temperature and Seebeck coefficient, is a fundamental principle in thermoelectricity. It has been regarded as an important application and direct experimental verification of the Onsager reciprocal relation (ORR) that is a cornerstone of irreversible thermodynamics. However, some critical questions still remain: (a) why Kelvin's proof-which omits all irreversibility within a thermoelectric transport process-can reach the correct result, (b) how to properly select the generalized-force-flux pairs for deriving the first Kelvin relation from the ORR, and (c) whether the first Kelvin relation is restricted by the requirement of the linear transport regime. The aim of the present work is to answer these questions based on the fundamental thermodynamic principles. Since the thermoelectric effects are reversible, we can redefine the Seebeck and Peltier coefficients using the quantities in reversible processes with no time derivative involved; these are renamed "reversible Seebeck and Peltier coefficients." The relation between them (called "the reversible reciprocal relation of thermoelectricity") is derived from the Maxwell relations, which can be reduced to the conventional Kelvin relation, when the local equilibrium assumption (LEA) is adopted. In this sense, the validity of the first Kelvin relation is guaranteed by the reversible thermodynamic principles and the LEA, without the requirement of the linear transport process. Additionally, the generalized force-flux pairs to obtain the first Kelvin relation from the ORR can be proper both mathematically and thermodynamically, only when they correspond to the conjugate-variable pairs of which Maxwell relations can yield the reversible reciprocal relation. The present theoretical framework can be further extended to other coupled phenomena.

Thermomass Theory in the Framework of GENERIC.

Thermomass theory was developed to deal with the non-Fourier heat conduction phenomena involving the influence of heat inertia. However, its structure, derived from an analogy to fluid mechanics, requires further mathematical verification. In this paper, General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework, which is a geometrical and mathematical structure in nonequilibrium thermodynamics, was employed to verify the thermomass theory. At first, the thermomass theory was introduced briefly; then, the GENERIC framework was applied in the thermomass gas system with state variables, thermomass gas density ρh and thermomass momentum mh, and the time evolution equations obtained from GENERIC framework were compared with those in thermomass theory. It was demonstrated that the equations generated by GENERIC theory were the same as the continuity and momentum equations in thermomass theory with proper potentials and eta-function. Thermomass theory gives a physical interpretation to the GENERIC theory in non-Fourier heat conduction phenomena. By combining these two theories, it was found that the Hamiltonian energy in reversible process and the dissipation potential in irreversible process could be unified into one formulation, i.e., the thermomass energy. Furthermore, via the framework of GENERIC, thermomass theory could be extended to involve more state variables, such as internal source term and distortion matrix term. Numerical simulations investigated the influences of the convective term and distortion matrix term in the equations. It was found that the convective term changed the shape of thermal energy distribution and enhanced the spreading behaviors of thermal energy. The distortion matrix implies the elasticity and viscosity of the thermomass gas.

Irreversibility and Action of the Heat Conduction Process.

Irreversibility (that is, the "one-sidedness" of time) of a physical process can be characterized by using Lyapunov functions in the modern theory of stability. In this theoretical framework, entropy and its production rate have been generally regarded as Lyapunov functions in order to measure the irreversibility of various physical processes. In fact, the Lyapunov function is not always unique. In the represent work, a rigorous proof is given that the entransy and its dissipation rate can also serve as Lyapunov functions associated with the irreversibility of the heat conduction process without the conversion between heat and work. In addition, the variation of the entransy dissipation rate can lead to Fourier's heat conduction law, while the entropy production rate cannot. This shows that the entransy dissipation rate, rather than the entropy production rate, is the unique action for the heat conduction process, and can be used to establish the finite element method for the approximate solution of heat conduction problems and the optimization of heat transfer processes.

The Principle of Least Action for Reversible Thermodynamic Processes and Cycles.

The principle of least action, which is usually applied to natural phenomena, can also be used in optimization problems with manual intervention. Following a brief introduction to the brachistochrone problem in classical mechanics, the principle of least action was applied to the optimization of reversible thermodynamic processes and cycles in this study. Analyses indicated that the entropy variation per unit of heat exchanged is the mode of action for reversible heat absorption or heat release processes. Minimizing this action led to the optimization of heat absorption or heat release processes, and the corresponding optimal path was the first or second half of a Carnot cycle. Finally, the action of an entire reversible thermodynamic cycle was determined as the sum of the actions of the heat absorption and release processes. Minimizing this action led to a Carnot cycle. This implies that the Carnot cycle can also be derived using the principle of least action derived from the entropy concept.

Reply to the Comments on: Tian Zhao et al. The Principle of Least Action for Reversible Thermodynamic Processes and Cycles. Entropy 2018, 20, 542.

The purpose of this reply is to provide a discussion and closure for the comment paper by Dr. Bormashenko on the present authors' article, which discussed the application of the principle of least action in reversible thermodynamic processes and cycles. Dr. Bormashenko's questions and misunderstandings are responded to, and the differences between the present authors' work and Lucia's are also presented.

Molecular Dynamics Studies on Thermal Transport Through Multiwalled Carbon Nanotubes.

The influence of the temperature and strength of the inter-wall interaction on the thermal conductivities of the (5,5) and (10, 10) double-walled carbon nanotubes (DWNTs) is studied by using molecular dynamics (MD) simulations with two different temperature control methods. One method is imposing heat baths (HBs) only on the outer wall, while the other is imposing HBs on both the two walls. The results show that the thermal conductivities of the DWNTs with the first method are about two-third of those with the second method. The relationship is the same even if the temperature and strength of the inter-wall interaction vary. Besides, the thermal conductivities of the DWNTs with the two different temperature control methods both slightly increase with the increasing energy parameters of Lennard-Jones (LJ) potential describing the inter-wall interaction and decrease with increasing temperature. Based on the analyses of the temperature profiles and phonon density of states (PDOS) spectra of the DWNTs with the two different temperature control methods, the results are well explained and the thermal transport mechanisms of multiwalled carbon nanotubes (MWNTs) under different conditions are explored.

Hydrodynamic Modeling of Heat Conduction in Nanoscale Systems.

Heat conduction in nanoscale systems has different behavior from bulk materials, which is applied to develop high performance thermoelectric material. The non-trivial behavior is caused by the ballistic-diffusive transport of heat carriers such as phonons. In this paper, we use the thermomass theory and phonon hydrodynamics model to establish a hydrodynamic model for phonon transport. In nanoscale systems, a Poiseuille flow of phonon gas is formed due to the boundary scattering. The thickness of boundary layer is proportional to the mean free paths of phonon. When the boundary layer thickness is comparable with the whole flow region, strong decrease of effective thermal conductivity happens. This method can serve as a fast evaluation method for nanoscale heat conduction.

General expression for entropy production in transport processes based on the thermomass model.

The entropy production in classical irreversible thermodynamics is expressed as a bilinear form of generalized (driving) forces and conjugated (driven) fluxes, which suffers from the arbitrary decomposition of the forces and the fluxes, and the possible negative entropy production in non-Fourier heat conduction problems (heat waves). This paper presents a general form of the entropy production for heat conduction based on the thermomass model, which is the product of the friction force and the drift velocity of the thermomass divided by the temperature; it holds true for both Fourier and non-Fourier heat conduction. Then a generalization of the entropy production is given for other kinds of linear and nonlinear transport processes. The general expression for entropy production is consistent with that given by extended irreversible thermodynamics, where the system entropy depends not only on the classical variables, but also on the dissipative fluxes, for example, the heat flux in heat conduction problems.

Thermal gradient induced actuation in double-walled carbon nanotubes.

Molecular dynamics simulations are applied to investigate the thermal gradient induced actuation in double-walled carbon nanotubes, where a temperature difference can actuate the relative motion of double-walled carbon nanotubes. The thermal driving force calculated through a stationary scheme is on the order of pico Newtons for a 1 K nm(-1) temperature gradient. The driving force is approximately proportional to the temperature gradient, but not sensitive to the system temperature. For the outer tube longer than 5 nm, the thermal driving force is nearly constant. For the outer tube shorter than 5 nm, however, the driving force decreases with decreasing tube length. The motion trace is found to depend on both the chirality pair and system temperature. A critical temperature can be defined by the potential barrier perpendicular to the minimum energy track of potential patterns. When the system temperature is higher than the critical temperature, the motion shows random behavior. When the system temperature is lower than the critical temperature, the motion, translational and/or rotational, is confined within the minimum energy track, which is indicative of the feasibility of directional control.

Molecular momentum transport at fluid-solid interfaces in MEMS/NEMS: a review.

This review is focused on molecular momentum transport at fluid-solid interfaces mainly related to microfluidics and nanofluidics in micro-/nano-electro-mechanical systems (MEMS/NEMS). This broad subject covers molecular dynamics behaviors, boundary conditions, molecular momentum accommodations, theoretical and phenomenological models in terms of gas-solid and liquid-solid interfaces affected by various physical factors, such as fluid and solid species, surface roughness, surface patterns, wettability, temperature, pressure, fluid viscosity and polarity. This review offers an overview of the major achievements, including experiments, theories and molecular dynamics simulations, in the field with particular emphasis on the effects on microfluidics and nanofluidics in nanoscience and nanotechnology. In Section 1 we present a brief introduction on the backgrounds, history and concepts. Sections 2 and 3 are focused on molecular momentum transport at gas-solid and liquid-solid interfaces, respectively. Summary and conclusions are finally presented in Section 4.

Silicon inhibition effects on the polymerase chain reaction: a real-time detection approach.

In the miniaturization of biochemical analysis systems, biocompatibility of the microfabricated material is a key feature to be considered. A clear insight into interactions between biological reagents and microchip materials will help to build more robust functional bio-microelectromechanical systems (BioMEMS). In the present work, a real-time polymerase chain reaction (PCR) assay was used to study the inhibition effects of silicon and native silicon oxide particles on Hepatitis B Virus (HBV) DNA PCR amplification. Silicon nanoparticles with different surface oxides were added into the PCR mixture to activate possible interactions between the silicon-related materials and the PCR reagents. Ratios of silicon nanoparticle surface area to PCR mixture volume (surface to volume ratio) varied from 4.7 to 235.5 mm2/microL. Using high speed centrifugation, the nanoparticles were pelleted to tube inner surfaces. Supernatant extracts were then used in subsequent PCR experiments. To test whether silicon materials participated in amplifications directly, in some cases, entire PCR mixture containing silicon nanoparticles were used in amplification. Fluorescence histories of PCR amplifications indicated that with the increase in surface to volume ratio, amplification efficiency decreased considerably, and within the studied ranges, the higher the particle surface oxidation, the stronger the silicon inhibition effects on PCR. Adsorption of Taq polymerase (not nucleic acid) on the silicon-related material surface was the primary cause of the inhibition phenomena and silicon did not participate in the amplification process directly.

Liquid flow in surface-nanostructured channels studied by molecular dynamics simulation.

Molecular dynamics simulations have been carried out to investigate the fluid wetting and flow in nanochannels whose surfaces are structured by an array of nanoscale triangular modules. We find that the surface nanostructures have a dual effect on the boundary slip and friction of the liquid nanoflow. On the one hand, the nanostructures can enhance the surface hydrophilicity for a hydrophilic liquid-solid interaction, and can increase the hydrophobicity for a hydrophobic interaction due to a nanoscale lotus effect. In particular, the nanostructured surface may show superhydrophobicity and lead to the large velocity slip of the liquid flow. On the other hand, simultaneously, the nanostructures distort the nanoscale streamlines of the liquid flow near the channel surfaces and block the nanoflow directly, which decreases the apparent slip length equivalently. The dual effect of the nanostructures on the surface wettability and the hydrodynamic disturbance results in a nonmonotonic dependence of the slip length on the nanostructure size. The simulations imply that the surface nanostructures can be applied to control the friction of liquid micro- and nanoflows.