On the structuring of thermodynamic fluxes: a direct implementation of the dissipation inequality

Consistency with the usual postulates of thermodynamics of irreversible processes is shown to lead to a particular structuring of any system of thermodynamic fluxes in terms of a dissipative part and a nondissipative part. Any system is thus shown to admit a generalized dissipation potential and to satisfy a system of symmetry relations.     

Asymptotic stability,Onsager fluxes and reaction kinetics

This paper constitutes a first step in the derivation of thermodynamics directly from the dynamics of physical systems. The existence of an asymptotically stable equilibrium point is used to construct a family of admissible entropy functions. These functions have nonnegative entropy production and assume an absolute maximum at the equilibrium point. A nonlinear generalization of the Onsager theory is then used to obtain a one-to-one correspondence between entropy production functions and the governing system of autonomous rate equations. The theory is applied to well stirred chemically reacting systems with constant temperature and pressure. This allows the derivation of chemical potentials, Gibbs' potentials and enthalpy for such systems. The rate equations for reaction kinetics and for classical thermodynamic reaction theory are obtained. Classic thermodynamic reaction theory is shown to give maximum entropy for constant enthalpy at the equilibrium point, while reaction kinetics gives this result only to within quadratic terms in the departure from equilibrium.     

Phenomenological derivation of the Onsager reciprocal relations

The Onsager reciprocal relation for a linear dissipative system with two fluxes is derived within the framework of a phenomenological approach using the property of nonnegative definiteness of the dissipative function. It is established that the Onsager relations are valid when the generalized fluxes are zero for nonzero generalized thermodynamic forces. In the general case, the Onsager reciprocal relations can be phenomenologically derived based on the theory of matrices and determinants or on the Prigogine principle of the entropy production minimum.     

Deformation and thermodynamic response for a crack dislocation model of brittle fracture

Concepts from dislocation and microcrack models are used to define idealized attributes describing the tip region of a crack. The attributes characterize both geometry and deformation; and this identifies a material defect species called a crack dislocation. With the attributes as variables, a scalar density function for each crack dislocation species can be defined.The crack dislocation is different from the common edge or screw species because it creates a new surface area as well as a displacement discontinuity as it propagates through a crystalline lattice. To model the discontinuities, a relative deformation functional is developed which depends on the crack dislocation density function. Since the crack dislocations are the only material defects assumed to contribute deformation discontinuities, the model is for brittle fracture only.The thermodynamic response uses primarily the methodology established by Gibbs. The existence of an internal energy functional is assumed. The methodology results in a definition for a thermodynamic potential for crack dislocation kinetics, a generalization of the Griffith crack propagation concept, and a local measure for the surface strain energy density changes on the crack dislocation line. The equilibrium thermodynamic potential for crack dislocation kinetics introduces a thermodynamic concept to demarcate crack dislocation density transitions from stationary to non-stationary; hence, it provides an incipient fracture criterion that has a thermodynamic basis. For nonequilibrium thermodynamics, the Onsager formalism is used to model the rates and fluxes of the thermodynamic functions.     

Thermodynamic costs of dynamic function in active soft matter

Living matter combines complex structures and dissipative processes to achieve dynamic functions that rely on material organization in space and time. In this Review, we discuss recent progress in creating synthetic material systems capable of four such functions–keeping time, powering motion, building structures, and making copies. Chemical oscillators coordinate the temporal activity of material assemblies; molecular motors and active colloids convert chemical energy into mechanical forces and motions; chemical activation of self-assembling components provides temporal control over dissipative structures; information-rich nanomaterials replicate their structures in exponential fashion. These and other dynamic functions cannot be achieved at thermodynamic equilibrium but instead require flows of energy and matter to create and maintain spatiotemporal order. Such systems are captured within the framework of stochastic thermodynamics, which describes the fluctuating thermodynamic quantities of driven systems. Even far from equilibrium, these quantities obey universal relations, which establish fundamental trade-offs between the rate of energy dissipation and performance metrics such as precision, efficiency, and speed. For each function considered, we present a simple kinetic model that offers general insights that inform the design and creation of dissipative material systems capable of dynamic functions. Overall, we aim to bridge experimental efforts in active soft matter and theoretical advances from stochastic thermodynamics to inform future research on material systems inspired by living matter.     

On the incorrectness of the traditional proof of the prigogine principle of minimum entropy production

It is demonstrated that the traditional proof of the Prigogine principle of minimum entropy production is incorrect. Provided that the Onsager reciprocal relations are satisfied, the Prigogine principle is valid when and only when generalized thermodynamic fluxes are simultaneously equal to zero at nonzero values of the generalized thermodynamic forces.     

Onsager reciprocity as a consequence of Maxwell reciprocity for “equidiffuse” fluids

Linear irreversible thermodynamics (LIT) principles are used to show, in the hypothetical “equidiffuse” case where all of the various phenomenological diffusion coefficients appearing in the linear near-equilibrium constitutive laws governing the diffuse transport of energy, multicomponent species, entropy, and volume (although not necessarily including momentum) are taken to be equal, that the Onsager reciprocal relations pertinent to nonequilibrium thermodynamics follow automatically from Maxwell’s reciprocal relations governing equilibrium thermodynamics. As such, this constitutes a purely macroscopic proof of Onsager reciprocity, the first of its kind. Although the equality of diffusion coefficients required in the equidiffuse limiting case does not represent a realistic possibility in the case of most mixtures, its adoption as a foil in uniting reversible and irreversible thermodynamics neither violates nor conflicts with any known physical law. Indeed, in some idealized sense such equality represents a perfection of the well-known analogy between these distinct physical transport phenomena. Moreover, as shown for mixtures of chemically similar dilute gases of comparable molecular weights (e.g., consecutive members of a homologous series) the equidiffuse assumption is generally quite good. As a bonus, our analysis — by virtue of its accord with all known precepts of macroscopic physics — constitutes a satisfactory resolution of the long-standing criticism of Coleman and Truesdell [B.D. Coleman, C. Truesdell, On the reciprocal relations of Onsager, J. Chem. Phys. 33 (1960) 28-31] centered on the possibility that Onsager symmetry might be nothing more than a tautology (derisively referred to in the literature as “Onsagerism”).     

Irreversible thermodynamics of nonlocal systems

Global laws of balance of momentum, moment of momentum and energy, together with local conservation of mass are reduced to point statements involving localization residuals. A fundamental functional inequality is then obtained which reduces to the Clausius-Duhem inequality for local theories. For nonlocal theories, the functional inequality states that the total internal production of heat of a material body at any given time is non-negative. This inequality is reformulated in terms of a functional inequality on a Hilbert space of ordered collections of L₂ functions. The general solution of this functional inequality is obtained and this leads to all admissible constitutive relations. The existence of dissipation potentials and symmetry relations are established for material bodies with admissible constitutive relations. Nonlocal analogues of Maxwell's reciprocity relations are also obtained as well as a proof of consistency with the results of thermostatics. Satisfaction of nonlinear forms of Onsager's reciprocity relations are shown to be equivalent to the requirement that the operators generating the admissible constitutive relations be potential operators. It is also shown the certain functionals of curves in a function space are odd under time reversal if and only if the nonlinear form of Onsager's reciprocity relations are satisfied. Thus, Gurtin's results [10] for processes which may be approximated by linear departures from equilibrium are extended to all processes with admissible constitutive relations. Similar results are established for significantly less restrictive sets of histories than those used by Gurtin and for a wide class of generalizations of the time reversal operator on such histories. Indications are given that satisfaction of invariance under superimposed rigid body motions implies satisfaction of the zero mean conditions for all localization residuals.     

Extended irreversible thermodynamics modeling for self-heating and dissipation in piezoelectric ceramics

Self-heating or dissipation of piezoelectric ceramic elements is observed to be severe under dynamic operations even in the linear range. In this paper, a nonequilibrium thermodynamic model is developed to delineate the coupled irreversible mechanical, electric, and thermal processes, which jointly contribute to dissipation. Specifically, additional nonequilibrium state variables, also known as thermodynamic fluxes, are brought in to describe each of these processes. The characteristic relaxation of these processes is modeled. The nonnegative rate of entropy production is found to be in quadratic form of thermodynamics fluxes. The energy balance equation, which governs the transformation between different energy forms, is obtained in the framework of extended irreversible thermodynamics. Using this model, the dissipation of a piezoceramic stack actuator under harmonic electric or mechanical loadings in linear operation range is studied. The harmonic-balance methods are utilized as solution techniques. In contrast to the existing piezoelectric dissipation models, the dissipation by the developed model is verified to nonlinearly depend on operating frequency, with a peak dissipation occurring at some operating frequency that is related to characteristic relaxation of irreversible processes. The measurements of newly introduced parameters are also discussed.     

Extended thermodynamics of mixtures of ideal gases

A phenomenological extended thermodynamic theory for a mixture of v fields is formulated. In this theory 13 v fields of partial mass densities, partial velocities, partial pressure tensors and partial heat fluxes are considered.The laws of Fourier, Navier-Stokes and Fick are obtained through an interation method akin to the Maxwellian iteration method in the kinetic theory of gases.It is shown also how the Onsager reciprocity relations in the presence of axial fields came out of this phenomenological theory without the use of statistical arguments.     

A general antisymmetry property of the constitutive equations of non-equilibrium thermodynamics

The constitutive equations of non-equilibrium thermodynamics which describe the exchange of heat, work and mass between a discontinuous thermodynamic system Σ and its environment Σ¹, show a general antisymmetry property. It is due to the fact that if the combined system Σ + Σ¹ forms an isolated thermodynamic system, the constitutive equations should be invariant against the substitution Σ Σ¹. This requirement is called the principle of environs invariance, some consequences of which are discussed especially in view of engineering applications.     

A thermodynamic analysis of rigid heat conductors

A thermodynamic approach to rigid heat conductors is proposed: it introduces the heat flux vector as independent variable while its temporal evolution is governed by a first order differential equation. The form of the second law is that proposed by Müller wherein the entropy flux and the entropy source are not given a priori but determined through constitutive equations. Restrictions on the constitutive equations are placed by the second law. Some properties, valid in the vicinity of equilibrium are established. In particular, it is shown that the present theory leads to a hyperbolic heat conduction equation, allowing for the propagation of heat as a thermal wave with a finite velocity. The concept of thermodynamic forces and fluxes is also introduced. The latter are seen to derive from a potential function plus an additional term. Finally, it is established under which conditions symmetry relations are satisfied.     

Green's functions for superfluid3He-A in the case of spin hydrodynamics: Influence of the spin-orbit coupling

The spin hydrodynamic equations for superfluid³He-A for the case of external, time-dependent fields are considered. The terms in equations containing these fields are found on the basis of a microscopic approach. Considering the linear response of the system to the switching on of external fields, formulas are found for suitable Green's functions constructed from operators entering the spin hydrodynamics. The Green's function connected with the order parameter operator has a 1/q ² singularity. Suitable connections between Green's functions lead to relations among kinetic coefficients (Onsager relations). A more detailed consideration of the spin-orbit coupling when the spin waves are not true Goldstone modes removes the mentioned singularity and the Onsager relations remain valid.     

Primitive thermodynamics: A new look at the Clausius-Duhem inequality

New answers to an old question in a new context provide the basis for constructing a thermodynamics which is surprisingly simple and yet capable of modeling viscous fluids, viscoelastic bodies, elastic bodies, and heat conduction. The results obtain from a complete solution of the reduced dissipation inequality without considering functional dependences on histories. Constitutive relations are shown to take the form of a sum of two collections of terms; the first collection being the quasistatic part of classic thermodynamics, and the second is the dynamic part. This dynamic part consists of a gradient of a dissipation potential with respect to the dynamic variables and a collection of terms which gives an identically zero supply of entropy. Absolute minimia of the dissipation potential with respect to the dynamic variables are shown to be attained only in thermostatic states. The theory possesses a full system of symmetry relations and consistency with thermostatics is demonstrated.     

On dissipative processes in Lorentzian manifolds

Use of the conditions that the total momentum-energy tensor by symmetric and have vanishing 4-divergence is shown to provide a framework for the construction of a Lorentz covariant thermodynamics of irreversible processes. Relativistically correct generalizations of recent work in irreversible thermodynamics are obtained, together with generalized systems of Onsager reciprocity relations. An elementary modification of the geometry and the field equations of general relativity is shown to provide one possible manner in which the dissipative effects of irreversible processes can be given a direct geometric interpretation.     

A creep model with damage based on internal variable theory and its fundamental properties

The creep damage is discussed within Rice irreversible internal state variable (ISV) thermodynamic theory. An ISV small-strain unified creep model with damage is derived by giving the complementary energy density function and kinetic equations of ISVs. The proposed model can describe viscoelasticity and, preferably, three phases of creep deformation. Creep strain results from internal structural adjustment, and different creep stages accompany different thermodynamic properties in terms of flow potential function and energy dissipation rate. During the viscoelastic process, the thermodynamic state of the material system tends to equilibrate spontaneously. The thermodynamic state of the material system without damage tends to equilibrate or achieve steady state after loading. Kinetic equations of ISVs can be derived by one single flow potential function, and the energy dissipation rate decreases monotonically over time. In the entire creep damage process, multiple potentials are needed to characterise evolution of ISVs, rotational fluxes are presented in affinity space, and the thermodynamic state of material system tends to depart from the steady or equilibrium state. The energy dissipation rate can be a measure of the distance between the current thermodynamic state and the equilibrium state. The time derivative of the rate can characterise the development trend of the material, and the integral value in the domain may be regarded as indices to evaluate the long-term stability of the structure.     

Thermodynamic properties of Tl5Se2Cl-based solid solutions

The equilibrium subsolidus phase diagram of the TlCl-Tl₂Se-TlSe system has been mapped out using X-ray diffraction analysis and emf measurements on thallium concentration cells. The Tl₅Se₂Cl compound has been shown to be a nonstoichiometric phase with a homogeneity region extending over a considerable part of the Tl₂Se-Tl₅Se₂Cl-Tl₅Se₃ composition triangle. The emf results have been used to evaluate the relative partial thermodynamic functions of the thallium in the alloys studied and the standard integral thermodynamic functions (ΔG ⁰(298 K), ΔH ⁰(298 K), and S ⁰(298 K)) of Tl₅Se₂Cl and Tl₅Se₂Cl-based solid solutions.     

On the problem of the phase relations in the B-BN system at high pressures and temperatures

Phase relations and chemical interaction in the B-BN system have been studied in situ using X-ray diffraction with synchrotron radiation at pressures to 5.3 GPa and temperatures to 2700 K. The results obtained have been used for the thermodynamic analysis of this system and for the construction of its phase diagram at 5 GPa. The values of thermodynamic functions of the phases competing in the B-BN system at high pressures and temperatures have been calculated in the framework of phenomenological models, unknown parameters of which have been found from the experimental data. It has been shown that only one thermodynamically stable boron subnitride (rhombohedral B₁₃N₂) exists in the system. It melts incongruently and forms an eutectic equilibrium with boron.