W. Muschik

Thermodynamics with Internal Variables. Part I. General Concepts

The first part of this review replaces the concept of internal variables of state in the general context of non-equilibrium thermodynamics. This offers the slightest deviation from the classical Theory of irreversible processes (TIP) while exhibiting a flexibility and versatility that other attempts at generalizing TIP cannot sustain. Here the basic working hypotheses on which the application of internal-variable theory relies, are specified. The use of TIP, after introduction of the appropriate axiom of local equilibrium state, or of a pseudopotentia l of dissipation is emphasized. The role of internal state variables in describing miscrostructures and the current misunderstanding with internal degrees of freedom are clarified. However, the resemblance or identity with order parameters of phase-transition theory as well as essential differences with extended thermodynamics are acknowledged. In all, internal variables epitomize the use of systems of evolution in mathematical continuum physics.

A sketch of continuum thermodynamics

Abstract Different formulations of non-equilibrium continuum thermodynamics are discussed: Thermodynamics of Irreversible Processes (TIP), Rational Thermodynamics (RAT), Extended Thermodynamics (ET), Mesoscopic Continuum Thermodynamics (MT), and the GENERIC version of thermodynamics. Concepts as constitutive quantity, state space, material frame indifference, exploiting dissipation inequality, mesoscopic variables, and GENERIC balance equations are taken into consideration.

Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation

Starting out with the global balance equations of mass, momentum, angular momentum, and energy formulated on the so-called ten-dimensional doubled phase of position, velocity, orientation, and orientation change velocity, the appropriate local balances are derived, which are defined on the five-dimensional half of the doubled phase space including time, position, and the microscopic director. These so-called orientation balance nematic liquid crystals whose alignment need not be uniform as it is presupposed in theories using macroscopic director fields. In R3 we get the usual phenomenological balance equations of micropolar media having the advantage that the balanced quantities are defined statistically. By expanding the orientation distribution function into a series of multipoles we get alignment tensor fields and an additional alignment tensor balance equation on R3.

Empirical foundation and axiomatic treatment of non-equilibrium temperature

SummaryOn the basis of well known empirems of classical thermodynamics, such as the First and Zeroth Laws, which lead to the definition of calorimeter systems by which heat exchanges can be measured, thermal interaction between thermally homogeneous systems is investigated by methods of set theory. In connexion with Carnaps axiom for a classical measuring quantity a thermodynamic analogue of the thermostatic temperature can be developed by generalized empirems which govern the thermal interaction between non-equilibrium and thermally homogeneous systems. An empirical identification of the contact temperature and its field formulation are given.

Thermodynamics with Internal Variables. Part II. Applications

The second part of this synthetic work presents and discusses the most spectacular and successful applications of the irreversible thermodynamics with internal variables. These include viscosity in both fluids and solids (in the former case, in complex fluids and structurally complex flows), viscoplasticity and rate-independent plasticity in small and finite strains, damage and cyclic plasticity, electric and magnetic relaxation, magnetic and electric hysteresis, normal and semi-conduction, superconductivity of deformable solids, and ferrofluids. In all cases the internal variables of interest are given some physical significance in terms of quantities defined at a sublevel of description. The relevant internal variables may be as varied as second-order tensors, real or complex valued scalars, and polar or axial vectors. Furthermore, the role played by internal variables in wave-propagation problems is emphasized through appropriate examples. The presentation ends with reaction-diffusion problems. This is illustrated by damage, plastic-strain localization and models for nerve-pulse dynamics. Globally, we have all ingredients of a truly post-Duhemian irreversible thermodynamics of complex behaviors.

Weakly Nonlocal And Nonlinear Heat Transport In Rigid Solids

A weakly nonlocal and nonlinear theory of heat conduction in rigid bodies is proposed. The constitutive equations generalize these of Fourier, Maxwell-Cattaneo and Guyer-Krumhansl. The proposed model uses the fundaments and the technique of extended irreversible thermodynamics. The main conclusion is that the presence of nonlocal terms in the transport equation for the heat flux implies a modification of the entropy flux; the latter is no longer given by its classical expression, i.e. the heat flux divided by the temperature, but contains extra contributions which are nonlinear in the heat flux and its gradient. These results arise as compatibility conditions with the second law of thermodynamics. A nonequilibrium temperature depending on the heat flux and generalizing the local equilibrium temperature is also emerging naturally from the formalism.

Orientation-balances for liquid crystals and their representation by alignment tensors

Abstract Up to now there are three different phenomenological concepts suitable to describe liquid crystals: The first one is the well known Ericksen-Leslie theory4 whose balance equations are formulated by use of a macroscopic director for needleshaped molecules with totally relaxed internal degrees of freedom. This macroscopic director is not well defined microscopically: It can not be identified with the orientation of a single molecule because the orientation of the molecules is statistically distributed. Therefore we need a distribution function for describing the orientation in the fluid, or we restrict ourselves to the case of total alignment. In that case the macroscopic director of the Ericksen-Leslie theory coincides with the common alignment of all molecules.

Endoreversible Thermodynamics: A Tool for Simulating and Comparing Processes of Discrete Systems

Abstract Endoreversible thermodynamics is concerned with reversible sub-systems that are in irreversible interaction with each other. Consequently, endoreversible thermodynamics represents the analogue for discrete systems to the local equilibrium hypothesis in continuum thermodynamics. Here a real cyclic 2-reservoir process is simulated by different endoreversible model processes. Simulation means that the simulating process has the same net heat exchanges, cycle time, power, entropy production, and efficiency as the original one. By introducing process-independent simulation parameters as constraints for the irreversible interaction, a family of comparative endoreversible processes is generated including the simulation of the original process. This procedure allows the process parameters of the family of comparative processes to be compared to those of the original one. The fraction “power of the real process over the maximal power inbetween the comparative family” is introduced as a parameter describing the process excellence.

A concept of non-equilibrium temperature

Abstract A thermodynamic analogue of the thermostatic temperature is defined by the vanishing heat exchange between a non-equilibrium system and a heat reservoir. Properties of this analogue called contact temperature are discussed making use of thermodynamic diagrams. Concerning the heat exchange these properties are similar to those of the thermostatic temperature.

Macroscopic constitutive equations for liquid crystals induced by their mesoscopic orientation distribution

Abstract By starting out with mesoscopic orientational balance equations for each orientational component of a liquid crystal which is described as a formal mixture, a set of independent macroscopic variables forming the state space is induced. This set includes a second-order tensorial measure of alignment, called the alignment tensor, and its derivatives. In terms of these state space variables constitutive equations are proposed by exploiting the dissipation inequality due to Coleman and Noll. The results are compared to those of Ericksen and Leslie, who described the alignment in a liquid crystal with only a macroscopic unit director field d ( x , t ) indicating the “mean orientation” of the media. In a recent paper Ericksen introduced beside the macroscopic director an additional scalar order parameter S( x ,t) and its derivatives (Maier-Saupe theory) which turns out to be the uniaxial case in the alignment tensor formulation. Also in this case the restrictions to the constitutive equations caused by the dissipation inequality are discussed.