Macroscopic Entropy of Non-Equilibrium Systems and Postulates of Extended Thermodynamics: Application to Transport Phenomena and Chemical Reactions in Nanoparticles

Abstract

In this work, we consider extended irreversible thermodynamics in assuming that the entropy density is a function of both common thermodynamic variables and their higher-order time derivatives. An expression for entropy production, and the linear phenomenological equations describing diffusion and chemical reactions, are found in the context of this approach. Solutions of the sets of linear equations with respect to fluxes and their higher-order time derivatives allow the coefficients of diffusion and reaction rate constants to be established as functions of size of the nanosystems in which these reactions occur. The Maxwell-Cattaneo and Jeffreys constitutive equations, as well as the higher-order constitutive equations, which describe the processes in reaction-diffusion systems, are obtained. by the parameters of atoms and molecules (e.g., for gases they are mean free path and mean time between two collisions of molecules, respectively). Macroscales is the characteristic macroscale of the system and characteristic time of changing of macroscopic parameters. In our case, spatial macroscale is a linear dimension of a system whether it is a particle size or a nanolayer thickness. When a spatial macroscale is much larger than the relevant microscale (at constant time scales), then local-equilibrium hypothesis remains valid that enables us to use classical irreversible thermodynamics. In turn, if a spatial macroscale is comparable or smaller than the microscale, there is no local equilibrium and the non-equilibrium system has to be described in the contents of more general theory.The latest development in irreversible thermodynamics led to elaboration of extended irreversible thermodynamics (EIT) which allowed study of fast and steep phenomena [11][12][13][14][15]. Extended irreversible thermodynamics involves investigations of irreversible processes with high frequencies and small length scales, which take place in nanosystems [16]. Heat conduction in thin layers [1,2], diffusion and chemical reactions in nanoparticles [17], as well as nanocatalysis [18,19] are of great interest in the context of EIT, where memory and non-local effects are taken into consideration [20].Extended irreversible thermodynamics is based on the generalization of classical local-equilibrium theory by using additional thermodynamic variables (or extra variables). Their choice is an important problem which can be solved by different ways. The most common route to extend a classical theory is the introduction of dissipative fluxes as extra variables [11][12][13][14]. For this kind of EIT, variables are both well-known fluxes, which arise in balance equations, and higher-order fluxes (fluxes of the fluxes) that allows the unlimited number of extra variables.Another type of EIT involves the time derivatives of common thermodynamic variables as extra variables [21][22][23][24][25]. In this case, the local-equilibrium hypothesis is not applicable, and we must proceed from a less strict postulate. Such a hypothesis is the local uniformity hypothesi...

Coexisting Ordered States, Local Equilibrium-like Domains, and Broken Ergodicity in a Non-turbulent Rayleigh-Bénard Convection at Steady-state

“…But first, we want to emphasize why this result is important. The local equilibrium condition until now has either been assumed as an a priori condition during the treatment of non-equilibrium steady-states, or it has been discussed in the literature entirely from either a pure theoretical or a phenomenological approach [17,46,47,48]. Moreover, some have even questioned the validity of the local equilibrium assumption [26,27].…”

Statement of the Second Law of Thermodynamics on the Basis of the Postulate of Nonequilibrium

“…As already noted, the main provisions defined by the second law of thermodynamics can be expressed in various formulations of this law in the form of postulates. One of these postulates is the postulate of the existence of thermodynamic equilibrium (Roh, 2014(Roh, , 2015Mishin, 2015;Skvortsov et al, 2016;Serdyukov, 2018). According to this postulate, any nonequilibrium isolated system over time comes to a state of thermodynamic equilibrium and cannot spontaneously get out of it.…”

Heat Transport on Ultrashort Time and Space Scales in Nanosized Systems: Diffusive or Wave-like?

“…The interest is largely motivated by technological needs such as thermal management in microelectronics and the ultra-fast laser processing of advanced metamaterials [1,4], i.e., artificial materials and media of designed properties, such as layered correlated materials [5,6]. Moreover, the heat transport at ultrashort space and time scales leads to unusual non-Fourier phenomena such as wavelike temperature propagation [4][5][6]12,13], size [14,15] and distance [15] dependent thermal conductivity, and boundary temperature jumps [1,14,15], which have raised an extensive body of literature concerning different conceptual questions of these phenomena [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. The problem is that when the characteristic length of the process is of the order of the mean free path (MFP) of energy carriers and/or the characteristic time scale of the process is of the order of the mean free time (MFT) of energy carriers, the thermal dynamics occur under far from local equilibrium conductions and cannot be described by classical Fourier law based on the local equilibrium assumption [31].…”

“…However, the HHCE cannot be used to describe heat conduction in nanosized systems where the nonlocal space effects [14][15][16][17][18]21,22,[26][27][28]31] and ballistic component of heat transport [14][15][16][17]19,23,24] lead to non-Fourier phenomena such as size-dependent thermal conductivity [1,9,[14][15][16][17] and boundary temperature jumps [14][15][16][17]19], which can be observed even in the steady-state [14][15][16][17]. Note that the HHCE in the steady-state reduces to the classical heat conduction equation of the parabolic type, and consequently, cannot describe these effects.…”

“…The nonlocal space effects [1,7,26,31] have been extensively studied in the context of heat conduction in nanosized systems [10,11,[14][15][16][17][18][21][22][23][24]28], in metals under ultrashort (pico-and femtosecond) laser irradiation [4,[31][32][33][34][38][39][40][41][42][43][44][45][46][47], and in biological [48][49][50] and heterogeneous systems [51][52][53]. Comprehensive reviews of different nonlocal heat conduction theories are presented in [41,42].…”