Irreversible thermodynamics linear theory

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

After the formulation of defect thermodynamics, it is necessary to understand the nature of rate constants and transport coefficients in order to make practical use of irreversible thermodynamics in solid state kinetics. Even the individual jump of a vacancy is a complicated many-body problem involving, in principle, the lattice dynamics of the whole crystal and the coupling with the motion of all other atomic structure elements. Predictions can be made by simulations, but the relevant methods (e.g., molecular dynamics, MD, calculations) can still be applied only in very simple situations. What are the limits of linear transport theory and under what conditions do the (local) rate constants and transport coefficients cease to be functions of state When do they begin to depend not only on local thermodynamic parameters, but on driving forces (potential gradients) as well Various relaxation processes give the answer to these questions and are treated in depth later. 

We will introduce basic kinetic concepts that are frequently used and illustrate them with pertinent examples. One of those concepts is the idea of dynamic equilibrium, as opposed to static (mechanical) equilibrium. Dynamic equilibrium at a phase boundary, for example, means that equal fluxes of particles are continuously crossing the boundary in both directions so that the (macroscopic) net flux is always zero. This concept enables us to understand the non-equilibrium state of a system as a monotonic deviation from the equilibrium state. Driven by the deviations from equilibrium of certain functions of state, a change in time for such a system can then be understood as the return to equilibrium. We can select these functions of state according to the imposed constraints. If the deviations from equilibrium are sufficiently small, the result falls within a linear theory of process rates. As long as the kinetic coefficients can be explained in terms of the dynamic equilibrium properties, the reaction rates are directly proportional to the deviations. The thermodynamic equilibrium state is chosen as the reference state in which the driving forces X, vanish, but not the random thermal motions of structure elements i. Therefore, systems which we wish to study kinetically must first be understood at equilibrium, where the SE fluxes vanish individually both in the interior of all phases and across phase boundaries. This concept will be worked out in Section 4.2.1 after fluxes of matter, charge, etc. have been introduced through the formalism of irreversible thermodynamics. 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

In the absence of gradients of salt concentration and temperature, flows of water and electric current in bentonite clay are coupled through a set of linear phenomenological equations, derived from the theory of irreversible thermodynamics (Katchalsky and Curran, 1967), making use of Onsager s Reciprocal Relations (Groenevelt, 1971) ... 

For dilute gas mixtures we may employ the linearity postulate in irreversible thermodynamics to obtain the transport fluxes for heat and mass. The fundamental theory is examined in chap 2 and we simply refer to the expressions (2.456) and (2.457). Moreover, a particular form of the generalized Maxwell-Stefan equations, i.e., deduced from (2.298) in chap 2, is given by ... 

We also note that the vector or tensor responses (3.187), (3.189) depend only on the vector or tensor driving forces respectively. This fact is known in linear irreversible thermodynamics as the Curie principle [36, 80, 88, 89] (cf. discussion in [34, 38]). Present theory shows however, that this property follows from the isotropy of constitutive functions and from the representation theorems of such linear functions, see Appendix A.2, Eqs.(A.ll)-(A.13) and (A.57)-(A.59). But representation theorems for nonlinear isotropic constitutive functions [64, 65] show that the Curie principle is not valid generally. 

Expressions (4.514), (4.515) are known as phenomenological equations of linear irreversible or non-equilibrium thermodynamics [1-5, 120, 130, 185-187], in this case for diffusion and heat fluxes, which represent the linearity postulate of this theory flows (ja, q) are proportional to driving forces (yp,T g) (irreversible thermodynamics studied also other phenomena, like chemical reactions, see, e.g. below (4.489)). Terms with phenomenological coefficients Lgp, Lgq, Lqg, Lqq, correspond to the transport phenomena of diffusion, Soret effect or thermodiffusion, Dtifour effect, heat conduction respectively, discussed more thoroughly below. 

R A Schapery, A theory of non-linear thermo-viscoelastidty based on irreversible thermodynamics . Proceedings 5th US National Congress Applied Mechanics, ASME, 1966, p 511. 

Due to their compactness and standard fabrication technology, the temperature in thermal flow sensors is often measured by thermocouples, which rely on the thermoelectric effect. The thermoelectric effect describes the coupling between the electrical and thermal currents, especially the occurrence of an electrical voltage due to a temperature difference between two material contacts, known as the Seebeck effect. In reverse, an electrical current can produce a heat flux or a cooling of a material contact, known as the Peltier effect. A third effect, the Thomson effect, is also connected with thermoelectricity, where an electric current flowing in a temperature gradient can absorb or release heat from or to the ambient [10, 11]. The relation between the first two effects can be described by methods of irreversible thermodynamics and the linear transport theory of Onsager in vector form. 

Clearly the first term in Eq. (5) is zero as the fluxes vanish when the thermodynamic forces are zero. The term which is linear in the forces is evidently derivable, at least formally, from the equilibrium properties of the system as the functional derivative of the fluxes with respect to the forces computed at equilibrium, X = 0. The quadratic term is related to what are known as the nonlinear contributions to the linear theory of irreversible thermodynamics. In general, Eq. (5) may be written as nonlinear functions of the forces in the expanded form... 

A phenomenological theory of ordinary diffusion in multicomponent sterns based on irreversible thermodynamics suggests that the ordinary diffusive flux of any species is a linear function of all the indepen nt composition gradients. In one dimension. 

All the relations in Sections B to F above may be regarded as originating in the Boltzmann superposition principle or the constitutive equation of Chapter 1, equation 7. The foundation of the theory has also been related to the principles of linear irreversible thermodynamics. It has been pointed out by Meixner that certain other postulates are taken for granted. Many of the specific predictions... 

Schapery, R.A., A Theory of Non-Linear Thermoviscoelasticity Based on Irreversible Thermodynamics , Proc. 5th U.S. National Congress of applied Mechanics, ASME, pp. 511-530, 1966. 

Heat capacity is the basic quantity derived from calorimetric measurements. For a full caloric description of a system, heat capacity information is combined with data on heats of transition, heats of reaction, etc., as outlined in Sect. 2.2.2. The basic descriptions of reversible and irreversible thermodynamics are given in Sects. 1.1.2, 2.1.1, and 2.1.2. In this section measurement and theory of heat capacity are discussed, leading to the /Advanced Tffermal y4nalysis System, ATHAS. This system was developed over the last 20 years to increase the precision of thermal analysis of linear macromolecules. It permits computation of the heat capacity from theoretical considerations or empirical addition schemes. Separating the heat capacity contribution fi-om the heat measured in a thermal analysis allows a more detailed interpretation of reversible and irreversible transitions and reactions. 

GK = Green-Kubo LIT = linear irreversible thermodynamics LRT = linear response theory NEMD = nonequilibrium molecular dynamics NESS = nonequilibrium steady state TTC = thermal transport coefficient TTCF = transient time correlation function. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

Abstract Based on the theory of irreversible process thermodynamics, non-linear stress-strain-temperature equations are derived, together with an expression for time-temperature equivalence. In addition, an equation of shift factor for time-temperature equivalence is also obtained. The parameters in the equations are experimentally determined and the main curves for creep compliance and cohesion of TOP granite are obtained by a series of creep tests. As a result, it is proved that both deformation and strength of the TOP granite follow the time-temperature equivalent principle. 

Electrophoresis and sedimentation potential also offer a test of predictions of thermodynamics of irreversible processes, provided these are supplemented by classical analysis of the data. Few measurements of sedimentation potential have been reported [1] and the theories due to Kruyt [2], Debye and Huckel [3] and Henry [4] are not in complete agreement. The thermodynamics of irreversible processes [5] may be helpful since the theory does not depend on any model. In the present chapter it is intended (i) to test linear phenomenological relations, (ii) to test the Onsager s reciprocal relation and (iii) to examine the validity of conflicting theories of electrophoresis. 

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