Phenomenological Transport Coefficients Research Articles

Dynamic density functional theory of polymers with salt in electric fields.

We present a dynamic density functional theory for modeling the effects of applied electric fields on the local structure of polymers with added salt (polymer electrolytes). Time-dependent equations for the local electrostatic potential and volume fractions of polymer, cation, and anion of added salt are developed using the principles of linear irreversible thermodynamics. For such a development, a field theoretic description of the free energy of polymer melts doped with salts is used, which captures the effects of local variations in the dielectric function. Connections of the dynamic density functional theory with experiments are established by relating the three phenomenological Onsager's transport coefficients of the theory to the mutual diffusion of electrolyte, ionic conductivity, and transference number of one of the ions. The theory is connected with a statistical mechanical model developed by Bearman and Kirkwood [J. Chem. Phys. 28, 136 (1958)] after relating the three transport coefficients to friction coefficients. The steady-state limit of the dynamic density functional theory is used to understand the effects of dielectric inhomogeneity on the phase separation in polymer electrolytes. The theory developed here provides not only a way to connect with experiments but also to develop multi-scale models for studying connections between local structure and ion transport in polymer electrolytes.

Soret separation and thermo-osmosis in porous media

When a temperature difference is applied over a porous medium soaked with a fluid mixture, two effects may be observed, a component separation (the Ludwig–Soret effect, thermodiffusion) and a pressure difference due to thermo-osmosis. In this work, we have studied both effects using non-equilibrium thermodynamics and molecular dynamics. We have derived expressions for the two characteristic parameters, the Soret coefficient and the thermo-osmotic coefficient in terms of phenomenological transport coefficients, and we show how they are related. Numerical values for these coefficients were obtained for a two-component fluid in a solid matrix where both fluid and solid are Lennard–Jones/spline particles. We found that both effects depend strongly on the porosity of the medium and weakly on the interactions between the fluid components and the matrix. The Soret coefficient depends strongly on whether the fluid is sampled from inside the porous medium or from bulk phases outside, which must be considered in experimental measurements using packed columns. If we use a methane/decane mixture in bulk as an example, our results for the Soret coefficient give that a temperature difference of 10 K will separate the mixture to about 49.5/50.5 and give no pressure difference. In a reservoir with 30% porosity, the separation will be 49.8/50.2, whereas the pressure difference will be about 15 bar. Thermo-osmotic pressures with this order or magnitude have been observed in frost-heave experiments.Graphic abstract

Relationships between chemical and tracer diffusion coefficients in strongly ionic crystals

Abstract In this paper, we make extensive use of various sum-rules relating the phenomenological transport coefficients in randomly mixed systems in order to address cation chemical interdiffusion and tracer diffusion in strongly ionic crystals such as alkali and silver halides and alkaline earth fluorides, oxides and silicates showing Schottky disorder. We derive general expressions for the intrinsic and inter-diffusivities for the case of unbound vacancies. Next, we derive a general expression relating the chemical interdiffusivity and the cation and anion tracer diffusivities. This expression reduces to the well-known Darken-Manning and Nernst-Planck equations according to the relative magnitude of the anion mobility. This analysis is repeated for the vacancy-pair mechanism. It is shown that the resulting expressions have exactly the same formal structure as for the unbound vacancies case, but the internal structure of the transport quantities differs because of the different correlation behaviour of the vacancy-pair mechanism compared with that of unbound vacancies.

Dissipative superfluid relativistic magnetohydrodynamics of a multicomponent fluid: The combined effect of particle diffusion and vortices

The authors formulate dissipative magnetohydrodynamic equations for finite temperature superfluid and superconducting mixtures, accounting for particle diffusion and the possible presence of vortices. They relate phenomenological transport coefficients to mutual friction and momentum transfer rates of the microscopic theory and suggest that this will find wide application in the study of neutron stars, particularly their magnetothermal evolution as well as their oscillations.

Erratum to: Electrokinetic Onsager coefficients and energy conversion in deformable nanofluidic channels

The original published article contains some errors which are corrected below. — Section 1, paragraph 2 is replaced by: “...where the Lij’s are called the Onsager phenomenological transport coefficients. Here, L11 is proportional to the permeability of the porous medium, L22 corresponds to the effective electric conductivity, and L21 and L12 are the coefficients for the streaming current and the electroosmotic flow, respectively. Note that the above formulation assumes a vanishing concentration of co-ions, which is the case that this contribution focuses on.”

The Vacancy-Wind Factor and the Manning Factor Occurring in Interdiffusion and Ionic Conductivity in Solids

In crystalline solids, during such processes as chemical interdiffusion in alloys, ionic conductivity and the annealing out of radiation damage there will inevitably be a net flux of vacancies. In most cases, when different species of atoms have different jump rates with vacancies within a net flux of vacancies, the phenomenon of the vacancy-wind effect will occur. This effect was first discovered in the 1960s by the late Dr John Manning. It is a subtle phenomenon that comes about because of the local redistribution of the equilibrium concentration of vacancies with respect to two or more species of drifting atoms in a driving force. The effect is captured in various ‘vacancy-wind factors’ (some of which are now sometimes called Manning factors) which formally arise from non-zero off-diagonal Onsager phenomenological transport coefficients and non-unity values of the tracer correlation factors. In this paper, the effect is reviewed and discussed.

Mechanical approach to chemical transport

Nonequilibrium thermodynamics describes the rates of transport phenomena with the aid of various thermodynamic forces, but often the phenomenological transport coefficients are not known, and the description is not easily connected with equilibrium relations. We present a simple and intuitive model to address these issues. Our model is based on Lagrangian dynamics for chemical systems with dissipation, so one may think of the model as physicochemical mechanics. Using one main equation, the model allows a systematic derivation of all transport and equilibrium equations, subject to the limitation that heat generated or absorbed in the system must be small for the model to be valid. A table with all major examples of transport and equilibrium processes described using physicochemical mechanics is given. In equilibrium, physicochemical mechanics reduces to standard thermodynamics and the Gibbs-Duhem relation, and we show that the First and Second Laws of thermodynamics are satisfied for our system plus bath model. Out of equilibrium, our model provides relationships between transport coefficients and describes system evolution in the presence of several simultaneous external fields. The model also leads to an extension of the Onsager-Casimir reciprocal relations for properties simultaneously transported by many components.

Generalized thermodynamic relations for a system experiencing heat and mass diffusion in the far-from-equilibrium realm based on steepest entropy ascent.

This paper presents a nonequilibrium thermodynamic model for the relaxation of a local, isolated system in nonequilibrium using the principle of steepest entropy ascent (SEA), which can be expressed as a variational principle in thermodynamic state space. The model is able to arrive at the Onsager relations for such a system. Since no assumption of local equilibrium is made, the conjugate fluxes and forces are intrinsic to the subspaces of the system's state space and are defined using the concepts of hypoequilibrium state and nonequilibrium intensive properties, which describe the nonmutual equilibrium status between subspaces of the thermodynamic state space. The Onsager relations are shown to be a thermodynamic kinematic feature of the system independent of the specific details of the micromechanical dynamics. Two kinds of relaxation processes are studied with different constraints (i.e., conservation laws) corresponding to heat and mass diffusion. Linear behavior in the near-equilibrium region as well as nonlinear behavior in the far-from-equilibrium region are discussed. Thermodynamic relations in the equilibrium and near-equilibrium realm, including the Gibbs relation, the Clausius inequality, and the Onsager relations, are generalized to the far-from-equilibrium realm. The variational principle in the space spanned by the intrinsic conjugate fluxes and forces is expressed via the quadratic dissipation potential. As an application, the model is applied to the heat and mass diffusion of a system represented by a single-particle ensemble, which can also be applied to a simple system of many particles. Phenomenological transport coefficients are also derived in the near-equilibrium realm.

On the correspondence between a large class of dynamical systems and stochastic processes described by the generalized Fokker–Planck equation with state-dependent diffusion and drift coefficients

In this paper using a projection approach and defining the adjoint-Lie time evolution of differential operators, that generalizes the ordinary time evolution of functions, we obtain a Fokker–Planck equation for the distribution function of a part of interest of a large class of dynamical systems. The main assumptions are the weak interaction between the part of interest and the rest of the system (typically non linear) and the average linear response to external perturbations of the irrelevant part. We do not use ad hoc statistical assumptions to introduce as given a priori phenomenological equilibrium or transport coefficients. The drift terms induced by the interaction with the irrelevant part is obtained with a procedure that is reminiscent of that developed some years ago by Bianucci and Grigolini (see for example (Bianucci et al 1995 Phys. Rev. E 51 3002)) to derive in a ‘genuine’ way thermodynamics and statistical mechanics of macroscopic variables of interest starting from microscopic dynamics. However here we stay in a more broad and formal context where the system of interest could be dissipative and the interaction between the two systems could be non Hamiltonian, thus the approach of the cited paper can not be used to obtain the diffusion part of the Fokker–Planck equation. To face the problem of dealing with the series of differential operators stemming from the projection approach applied to this general case, we introduce the formalism of the Lie derivative and the corresponding adjoint-Lie time evolution of differential operators. In this theoretical framework we are able to obtain well defined analytic functions both for the drift and the diffusion coefficients of the Fokker–Planck equation. We think that the basic elements of Lie algebra introduced in our projection approach can be useful to achieve even more general and more formally elegant results than those here presented. Thus we consider this paper as a first step of this formal path to statistical mechanics of complex systems.

Linear response and hydrodynamics for granular fluids

A formal derivation of linear hydrodynamics for a granular fluid is given. The linear response to small spatial perturbations of a homogeneous reference state is studied in detail, using methods of nonequilibrium statistical mechanics. A transport matrix for macroscopic excitations in the fluid is defined in terms of the response functions. An expansion in the wave vector to second order allows identification of all phenomenological susceptibilities and transport coefficients through Navier-Stokes order in terms of appropriate time correlation functions. The transport coefficients in this representation are the generalization to granular fluids of the familiar Helfand and Green-Kubo relations for normal fluids. The analysis applies to a variety of collision rules. Important differences in both the analysis and results from those for normal fluids are identified and discussed. A scaling limit is described corresponding to the conditions under which idealized inelastic hard sphere models can apply. Further details and interpretation are provided in the paper following this one, by specialization to the case of smooth, inelastic hard spheres with constant coefficient of restitution.

From the Atomic Jump Frequencies to the Phenomenological Transport Coefficients

The SCMF theory based on an atomic model of atom‐vacancy exchange frequencies yields general expressions of the phenomenological coefficients Lij of a multi‐component alloy with any crystallographic structure. The limitations and future improvements of the Self‐Consistent Mean‐Field (SCMF) approach are easily related to the statistical approximation of the thermodynamic correlations and to the time‐dependent effective interactions used to describe the kinetic correlations induced by the vacancy diffusion mechanism.

Six-Jump-Cycle Mechanism for Collective Correlations in Nonstoichiometric Intermetallic Compounds

The six-jump-cycle (6JC) mechanism is used to derive expressions for collective correlation factors in a nonstoichiometric binary intermetallic compound AB. The 6JC is used as a fundamental unit for the cycle involving a perfectly ordered configuration and a two-jumpcycle (2JC) as a fundamental unit for the cycle involving existing antistructural atoms. The jump frequency for the 6JC is calculated in terms of a four-frequency-model using the mean first passage concept of Arita et al., while the jump frequency for the 2JC is taken to be the harmonic mean of the individual jump frequencies. The expressions for phenomenological transport coefficients are obtained through the linear response approximation using the kinetic equation approach. The results for collective correlation factors are compared with Monte Carlo simulation and are found to be in reasonably good agreement when the ratio of jump frequencies of regular site and antistructural atoms is of the order of 10-1.

Calculation of Phenomenological Coefficients by Monte Carlo Computer Simulation Methods

In this paper we first review the principal indirect and direct Monte Carlo methods for calculating the Onsager phenomenological transport coefficients in solid state diffusion. We propose a new Monte Carlo method that makes use of a steady state calculation of a flux of atoms that is driven by a difference in chemical potential of the atoms between a source and a sink plane. The method is demonstrated for the simple cubic one component lattice gas with nearest neighbour interactions. The new method gives results in good agreement with a Monte Carlo method based on Einsteinian expressions for the phenomenological coefficients.

Collective Correlation Factors in Random Non-Stoichiometric Inermetallic Compounds

In this paper Manning random alloy model has been extended to the binary nonstoichiometric intermetallic compound of the B2 structure. Two sub-lattices, that are dynamically independent in six-jump cycle (6JC) mechanism, are coupled together by taking into consideration the vacancy motion as a sequence of nearest neighbour jumps in random directions. The linear response expressions for the phenomenological transport coefficients are evaluated making use of the kinetic equation approach. The expressions for collective correlation factors are derived in terms of the equilibrium partial atomic concentrations and jump frequencies. Results are compared with Monte Carlo simulation results using the four-frequency model.

Sum-rule relationships among phenomenological coefficients: Simplifications for the analysis of segregation and chemical diffusion

In this review article, the authors first introduce the phenomenological transport coefficients, and the relationships between them and the tracer diffusion coefficients. Next, the authors discuss a sum rule relating phenomenological coefficients themselves in the random-alloy model. The authors then consider several applications of the sum rule to diffusion problems. These applications include intrinsic diffusion in multicomponent alloys, chemical diffusion in strongly ionic mixed cation crystals, and the segregation (demixing) of cations in mixed oxides in an oxygen potential gradient and/or an electric field. In each case, a substantial simplification is possible as a result of the sum rule.

Relationships between chemical and tracer diffusion coefficients in strongly ionic crystals

Abstract In this paper, we make extensive use of various sum-rules relating the phenomenological transport coefficients in randomly mixed systems in order to address cation chemical interdiffusion and tracer diffusion in strongly ionic crystals such as alkali and silver halides and alkaline earth fluorides, oxides and silicates showing Schottky disorder. We derive general expressions for the intrinsic and inter-diffusivities for the case of unbound vacancies. Next, we derive a general expression relating the chemical interdiffusivity and the cation and anion tracer diffusivities. This expression reduces to the well-known Darken-Manning and Nernst-Planck equations according to the relative magnitude of the anion mobility. This analysis is repeated for the vacancy-pair mechanism. It is shown that the resulting expressions have exactly the same formal structure as for the unbound vacancies case, but the internal structure of the transport quantities differs because of the different correlation behaviour of t...

Prediction of high temperature demixing in oxides in an electric field or an oxygen potential gradient

During exposure to an oxygen potential gradient or an electric field at high temperatures, semi-conducting and initially homogeneous mixed ceramic oxides of the rock-salt type show demixing of the cations. The main assumption made in previous theoretical treatments of demixing is that the off-diagonal phenomenological transport coefficients can be ignored. In this paper, off-diagonal coefficients are calculated by making use of the exact sum-rule expression relating the phenomenological transport coefficients. The demixing analysis is then greatly simplified and made exact. The analysis is verified with Monte Carlo computer simulation. Results are presented for steady state composition profiles in oxygen potential gradients and electric fields. Unusual steady state demixing composition profiles are possible in the quaternary oxide (A,B,C)O. Demixing profiles in (Co,Mg)O (oxygen potential gradient) and (Co,Ni)O (electric field) are analysed.

Analysis of kinetic demixing of (A,B)O oxides in an electric field and an oxygen potential gradient

Steady-state kinetic demixing in an electric field and an oxygen potential gradient (vacancy composition gradient) of semiconducting and initially homogeneous mixed oxides (A,B)O of the rock salt type is analysed. The usual assumption of neglecting the off-diagonal phenomenological coefficients is avoided by assuming a random distribution of cations and making use of the exact sum-rule expression that relates the phenomenological transport coefficients for this situation. The kinetic demixing analysis is then considerably simplified. We verify the analysis with Monte Carlo computer simulation. It is found that for demixing in an electric field, the vacancy composition profile is not in fact flat, but exhibits a maximum that depends on the ratio of the cation-vacancy exchange frequencies. When a vacancy composition gradient and an electric field are both applied, it is shown that unusual steady state vacancy composition profiles can result.

The sum-rule relation among phenomenological transport coefficients and its consequences in the analysis of collective diffusion problems

In this paper, we discuss the sum-rule that relates the phenomenological coefficients in the multicomponent random alloy system in the case where the isolated vacancy mechanism is operating. We present applications of this sum-rule to intrinsic diffusion in multicomponent alloys, intrinsic diffusion in mixed alkali halides and demixing of multicomponent transition metal oxides in an oxygen potential gradient and an electric field. In each case, a very substantial simplification in the analysis is made possible because of the sum-rule.

A Formalism to Describe Demixing of Mixed Oxides of Large and Small Grain Size in an Electric Field

AbstractIn this paper, we address the theory of electric field demixing of the cations in semi-conducting and initially homogeneous mixed ceramic oxides. We include the off-diagonal phenomenological transport coefficients by assuming a random distribution of cations and making use of an exact sum-rule expression. The steady state atomic composition profiles are shown to be quite different when different assumptions are made about vacancy equilibration. We show that in the case of large grained material (where vacancies in the interior do not equilibrate easily to the external oxygen partial pressure), a maximum develops in the steady state vacancy composition profiles. This behaviour is verified by independent Monte Carlo simulations.