ON MAGNETIC RELAXATION EQUATION FOR ISOTROPIC MAGNETIZABLE REACTING FLUID MIXTURES

In this paper a theory for magnetic relaxation phenomena developed in the framework of thermodynamics of irreversible processes (TIP) with internal variables by G. A. Kluitenberg (G.A.K.) and the author is reviewed. Analogies and correlations with thermodynamic theories with internal variables for mechanical distortional phenomena and dielectric relaxation phenomena, derived, respectively, by G. A. K., and by G. A. K. and the author, following the same procedure, are put forward as evidence. It is assumed that if n different types of irreversible microscopic phenomena give rise to magnetic relaxation, it is possible to describe these microscopic phenomena splitting the total specific magnetization in n + 1 parts and introducing n of these partial specific magnetizations as internal variables in the thermodynamic state space. The phenomenological equations are formulated in the anisotropic and isotropic cases. Finally, linearizing the equations of state, generalizations of the Snoek equation for magnetic relaxation phenomena are obtained by eliminating the internal variables. Some results obtained in the paper are new.

Using the methods of classical irreversible thermodynamics with internal variables, the heat dissipation function for magnetizable ani­sotropic media, in which phenomena of magnetic relaxation occur, is derived. It is assumed that if different types of irreversible microscopic phenomena give rise to magnetic relaxation, it is possible to describe these microscopic phenomena splitting the total specific magnetization in two irreversible parts and introducing one of these partial specific magnetizations as internal variable in the thermodynamic state space. It is seen that, when the theory is linearized, the heat dissipation func­tion is due to the electric conduction, magnetic relaxation, viscous, magnetic irreversible phenomena. This is the case of complex media, where different kinds of molecules have different magnetic susceptibili­ties and relaxation times, present magnetic relaxation phenomena and contribute to the total magnetization. These situations arise in nuclear magnetic resonance in medicine and biology and in other fields of the applied sciences. Also, the heat conduction equation for these media is worked out and the special cases of anisotropic Snoek media and anisotropic De-Groot-Mazur media are treated.

On vectorial internal variables and dielectric and magnetic relaxation phenomena

Using the general methods of non-equilibrium thermodynamics, a theory for anisotropic magnetizable media in which magnetic relaxation phenomena occur is formulated. In this paper, some results are revised, some others are new. First, a critical revision is given in the case where it is assumed that n microscopic phenomena give rise to magnetic relaxation, and the contributions of these phenomena to the macroscopic magnetization are introduced as internal variables in the Gibbs relation. Phenomenological equations and linear state equations are formulated, and magnetic relaxation equations generalizing Snoek equation are obtained. Then, new results are derived in the special case where all cross-effects are neglected, except for possible effects among the different types of magnetic relaxation phenomena, and by direct computations, generalized Snoek equations are worked out when the magnetization axial vector is additively composed of two irreversible parts, and in the case of anisotropic Snoek media and anisotropic De Groot–Mazur media. Also, particular results are presented and reviewed in the cases where the considered media have symmetry properties, under orthogonal transformations, which are i) invariant with respect to all the rotations of the frame of axes; ii) invariant with respect to all the rotations and inversions of the frame of axes.

On transformations of internal vectorial variables in the thermodynamic theory of dielectric relaxation and the invariance of Onsager's reciprocal relations

In some previous papers, within the framework of the thermody namics of irreversible processes with internal variables, a linear theory for magnetic relaxation phenomena in anisotropic mixtures, consisting of n reacting fluid components, was developed. In particular, assuming that the macroscopic magnetization m can be split in two irreversible parts m = m(0) + m(1) a generalized Snoek equation was derived. In this paper we derive for these reacting anisotropic mixtures the heat conduction equation. We show that the heat dissipation function is due to the chemical reactions, the magnetic relaxation, the electric conduc tion, the viscous, magnetic, temperature fields and the diffusion and the concentrations of the n fluid components. Also, the Snoek and De Groot special cases are studied. The obtained results find applications in nuclear resonance, in biology, in medicine and other fields, where different species of molecules have different magnetic susceptibilities and relaxation times and contribute to the total magnetization.

A model for elastic-plastic hardening materials with coupled damage based on internal variables is described. Each kinematic variable is partitioned in the sum of a reversible and an irreversible part. The first enters as argument of the free energy, the other enters as argument of a dissipation functional D. that generates a generalised elastic domain K for the thermodynamic variables as the level set at 1 of the dissipation function. Following a general procedure for multivocal potentials, a family of variational principles can be obtained, whose stationary points represent the solution of the structural problem. 1. Aims and Limitations of the work It is presented a new phenomenological model for a class of elasticplastic-damaging materials, based on an internal variable description of the modifications of the microstructure. No attempt is made to give any physical interpretation for the internal variables. A consistent thermodynamic development of the basic equations is presented, based on the introduction of two functional ruling the reversible and the irreversible phenomena. Mild regularity hypotheses are made on these functional, that guarantee the mathematical treatment of the equations with the tools of convex analysis. The most significant point of the model is the consistent derivation of a (convex) elastic region from a dissipation functional of the kinematic variables (internal and external). In this way it is obtained a single elastic domain in the extended space of stresses and thermodynamic forces dual of the hardening and damage internal variables -, for which Transactions on Engineering Sciences vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533 98 Damage and Fracture Mechanics normality rule holds naturally. According to the choice of the dissipation functional, and therefore of the generalised elastic domain, coupling between plasticity and damage can be easily modelled. Various kinds of behaviour can be recovered, as the decrease of the yield strength or the degradation of the hardening modulus. The different damages are clearly distinguished in the model according to whether they influence the reversible or the irreversible part of the constitutive equations. Principal aim of the model is to use it in developing variational principles amenable for simple numerical treatment. It is shown that it is possible to obtain a series of energy functional as straightforward generalisation of those valid in the field of plasticity with internal variables. Particular attention is devoted to mixed principles, that the authors have shown in previous works to be particularly useful in the numerical analysis (Cuomo ). The paper will present the main features of the model, while the consequences of particular choices of the form of the governing potentials will not be discussed. The identification of the model parameters is not faced in the present work 2. State variables and ambient spaces Kinematic internal variables are introduced for describing both material hardening and damage, defined in suitable vector spaces. Conjugate thermodynamic variables are defined in adjoint spaces. Each kinematic variables is partitioned as the sum of a reversible and an irreversible part. The first enter as argument in the Helmoltz free energy F, that rules the generalised elastic (reversible) behaviour. The irreversible parts enter as arguments in a dissipation functional D, that rules the dissipating phenomena. Schematically it is: £ = £^,+£ E%> deformation space a = «^+cx^ e9 internal strain space co = (0 v (4) Linearity assumptions have been introduced, so small deformation only are considered. The operator C accounts for the possibility that internal microdeformations result into overall deformations, Owen; v is an external damage potential. In most of the classic works on damage and hardening it is assumed that d, C? reduce to the null operator. From the virtual power equality the equilibrium conditions follow in the usual way, Cc+C|'x=/ Ci'C = f (5) The second equation is non classical; in it t are external source of damage work. This contribution has been examined by Fremond 4. Constitutive equations 4.1 Free energy Many different forms of the free energy have been assumed in the literature. Following the pioneering works of Kachanov it is often introduced a damaging effect on the stored elastic energy of the type Transactions on Engineering Sciences vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533

In this paper a linear theory for dielectric relaxation phenomena in polarizable reacting fluid mixtures is developed, in the frame of thermodynamics of irreversible processes with internal variables. The microscopic irreversible phenomena giving rise to dielectric relaxation are described splitting the total specific polarization in two irreversible parts and introducing one of these partial specific polarizations as internal variable in the thermodynamic state vector. The phenomenological equations for these fluid mixtures are derived and, in the linear case, a generalized Debye equation for dielectric relaxation phenomena is derived. Special cases are also treated. Linear theories for polarizable continuous media with dielectric relaxation phenomena were derived in the same frame of non-equilibrium thermodynamics with internal variables in previous papers by one of the authors (LR). A phenomenological theory for these phenomena was developed by Maugin for complex materials, using microscopic considerations and introducing articular partial polarizations per unit mass. The obtained results in this paper have applications in several fields of applied sciences, as, for instance, in medicine and biology, where complex fluids presenting dielectric relaxation, are constitued by different types of molecules, with own dielectric susceptibility and relaxation time.

Multi-exponential decay is prevalent in magnetic resonance spectroscopy, relaxation, and imaging. This paper describes simple MATLAB and Python functions and scripts for regularized multi-exponential analysis methods for 1D and 2D data and example test problems and experiments. Regularized least-squares solutions provide production-quality outputs with robust stopping rules in ~5 and ~20 lines of code for 1D and 2D inversions, respectively. The software provides an open-architecture simple solution for transforming exponential decay data to the distribution of their decay lifetimes. Examples from magnetic resonance relaxation of a complex fluid, a Danish North Sea crude oil, and fluid mixtures in porous materials-brine/crude oil mixture in North Sea reservoir chalk-are presented. Developed codes may be incorporated in other software or directly used by other researchers, in magnetic resonance relaxation, diffusion, and imaging or other physical phenomena that require multi-exponential analysis.

The aim of our work was to find an unambiguous connection between irreversible macroscopic dynamics and reversible microdynamics that makes it possible to manifest irreversibility on a submacroscopic level without the use of coarse graining arguments. On this level the state of a physical system can be described by a field amplitude Ψ and the time evolution of this system is determined by a field equation for Ψ. For conservative systems, this field equation is formally identical with the linear Schrödinger equation, which can be constructed with the help of the classic Hamiltonian function for the corresponding problem. Regarding irreversible phenomena like damping due to a frictional force, for those dissipative systems no classic Hamiltonian function exists. Therefore the corresponding field equation cannot be obtained in the usual way. Nevertheless, also for dissipative systems it is possible to obtain a field equation in an unambiguous way using only Newton's form of classic mechanics. The result of our method is a nonlinear Schrödinger‐type field equation with a logarithmic nonlinearity. We discuss in more detail the properties of our logarithmic nonlinearity that corresponds to a macroscopic frictional force in a unique way. A figurative interpretation in terms of environment and interaction can be given. From a more formal point of view, the compatibility of our nonlinear operator with principles known from the theory of linear operators is investigated. One of the surprising results is the fact that although our nonlinear Hamiltonian HNL is “Hermitean” in the usual sense, in contrast to the linear theory an operator exp(i · HNL) is not unitary. Furthermore, in our theory the time‐derivative of the mean value of an operator is not only essentially determined by (the mean value of) its commutator with the Hamiltonian. There also occurs an additional term that causes irreversibility of the evolution and is connected with the feature of our theory that (in general) time derivative and construction of the mean value are no longer commuting operations. This fact shows some similarity with coarse graining theories, but in our theory the reason can be traced back unambiguously to an irreversible physical phenomenon.

The aim of our work was to find an unambiguous connection between irreversible macroscopic dynamics and reversible microdynamics that makes it possible to manifest irreversibility on a submacroscopic level without the use of coarse graining arguments. On this level the state of a physical system can be described by a field amplitude Ψ and the time evolution of this system is determined by a field equation for Ψ. For conservative systems, this field equation is formally identical with the linear Schrodinger equation, which can be constructed with the help of the classic Hamiltonian function for the corresponding problem. Regarding irreversible phenomena like damping due to a frictional force, for those dissipative systems no classic Hamiltonian function exists. Therefore the corresponding field equation cannot be obtained in the usual way. Nevertheless, also for dissipative systems it is possible to obtain a field equation in an unambiguous way using only Newton's form of classic mechanics. The result of our method is a nonlinear Schrodinger-type field equation with a logarithmic nonlinearity. We discuss in more detail the properties of our logarithmic nonlinearity that corresponds to a macroscopic frictional force in a unique way. A figurative interpretation in terms of environment and interaction can be given. From a more formal point of view, the compatibility of our nonlinear operator with principles known from the theory of linear operators is investigated. One of the surprising results is the fact that although our nonlinear Hamiltonian HNL is “Hermitean” in the usual sense, in contrast to the linear theory an operator exp(i · HNL) is not unitary. Furthermore, in our theory the time-derivative of the mean value of an operator is not only essentially determined by (the mean value of) its commutator with the Hamiltonian. There also occurs an additional term that causes irreversibility of the evolution and is connected with the feature of our theory that (in general) time derivative and construction of the mean value are no longer commuting operations. This fact shows some similarity with coarse graining theories, but in our theory the reason can be traced back unambiguously to an irreversible physical phenomenon.

Radiopharmaceutical enhancement by drug delivery systems: A review

In a previous paper, in the framework of extended irreversible thermodynamics with internal variables, a model for nanostructures with thin porous channels filled by a fluid flow was developed. The porous defects of porous channels inside the structure, described by a permeability structure tensor, may have a strong influence on the effective thermal conductivity, and their own dynamics may couple in relevant way to the heat flux dynamics. Here, in the linear case a generalized telegraph heat equation for thermal perturbations with finite velocity is derived in the anisotropic and isotropic case for the nanosystems taken into account. The thermal disturbances are so fast that their frequency becomes of the order of reciprocal of the relaxation time, given, for instance, by the collision time of heat carriers. Furthermore, the complete system of equations describing the behaviour of the media under consideration is worked out and discussed. The obtained results have applications in defect engineering and an important technological interest.

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.

Two entropy functions are currently in use: the thermostatic entropy, defined by Caratheodory's theory, and the thermodynamic entropy, defined by the theory of irreversible processes. Both entropy concepts are confined to systems without internal variables, and both can be shown to be equal by substituting the respective balance of internal energy to which they are related by the integrating factor, 1/Θ. When irreversible internal phenomena are present, represented by internal coordinates and their conjugate affinities, they become part of the entropy production, but not of the energy balance, and the two entropies are no longer equal. It has been shown in the literature that by multiplication by a second integrating factor, an extended entropy function for systems with internal variables can be derived. It is the purpose of this paper to present a method for the determination of this integrating factor. Under certain conditions, the latter may be unity; such is shown to be the case with the Gibbs equation for gas mixtures.