Bogusław Bożek

Interdiffusion: compatibility of Darken and Onsager formalisms

Abstract A unified treatment of interdiffusion in various reference frames is presented in view of the consistency between Nernst–Planck, Onsager and Darken formalisms. A present discussion involves (i) material, (ii) laboratory and (iii) Rth component reference frames. It is shown that when the fluxes and forces are linearly interrelated, one can define a symmetric matrix of phenomenological coefficients. Its explicit form is derived. Problem is generalised to the case of different molar volumes that satisfy Vegard law, which presents entirely new result. An evidence is given that the Onsager fluxes can be equivalently treated as the fluxes due to the material drift and diffusion, as accounted in Darken treatment. The entropy production is the same within both formalisms.

Electro-Diffusion in Multicomponent Ion-Selective Membranes; Numerical Solution of the Coupled Nernst–Planck–Poisson Equations

Numerical simulations of evolution of the potentials and impedance spectra of ionselective membranes (ISEs) with ionic sites are presented. The Nernst–Planck–Poisson and continuity equations (NPP) are solved numerically by means of the finite difference method, the Rosenbrock solver and with the use of Matlab platform. Transient solutions for ion-selective electrodes under open- and closed-circuit conditions are computed. The potential-time response to small-current perturbation is used for determination of complex impedances. We present simulations of ISEs as a function of varying diffusivities and ionic concentrations in the “bathing” solutions at interfaces. It is shown that the non-Nernstian behavior of passive membrane electrodes is a result of kinetic constraints at the interfaces, which is manifested in the appearance of an additional arc between the high-frequency bulk and the low-frequency (Warburg) arcs. The presented approach directly relates the diffusivities in the membrane and the interface properties (heterogeneous rate constants determining the transport across interfaces) to the characteristic features of impedance spectra (dimensions and characteristic radial frequencies). NPP problem solved on the Matlab platform allows simulating of the non-linear effects in electro-diffusion.

Bi-velocity model of mass transport in two-phase zone of ternary system

Bi-velocity (Darken) method, which includes material drift and diffusion fluxes, is for the first time applied to describe mass transport in multiphase materials. The respective model is formulated in details and then applied for a ternary diffusion couple in which a two-phase zone can grow during isothermal diffusion. Thanks to the use of a phase-field order parameter, identified here with volume fraction of a chosen phase, the mass transport throughout both phases within a two-phase zone can be considered. The model allows smooth crossing of the type 1 boundary, which makes the mass transport equations valid in both single- and two-phase regions. The solution obtained for 1D geometry provides: (1) a diffusion path in the concentration triangle, (2) element-concentration profiles, (3) volume fractions of the phases in the two-phase zone and (4) drift-velocity distribution along x axis parallel to the mass transport. As an example, the interdiffusion in the diffusion couple of Type 0 boundary is modelled. The zigzag diffusion path is predicted and the profiles of the element concentrations are simulated. For the first time, the drift velocity for the diffusion in two-phase system is determined and correlated with the changes in volume fractions of the phases.

On the Consistency of the Darken Method with the Onsager Representation for Diffusion in Multicomponent Systems

A consistency between the Darken method and the Onsager representation for cross diffusion in multicomponent system is shown. The justification is made by defining new sets of forces and fluxes linearly interrelated by a symmetric matrix of phenomenological coefficients. For the first time, the system of the components having various molar volumes is treated in this way. It is shown that the transformation leaves the entropy production unchanged. As an example, the entropy production for interdiffusion in the ternary Co-Fe-Ni diffusion couple is calculated and compared with mixing entropy.

Interdiffusion: Consistency of Darken's and Onsager's Methods

The Nernst-Planck flux formula is used in Darkens method to obtain the interdiffusion fluxes. The effective interdiffusion potentials, derived for the independent components in the system, allow obtaining the symmetrical matrix of the interdiffusion coefficients. The transport coefficients for 2, 3 and r-component system are presented. Interpretation of obtained matrixes in the light of Onsagers theory of irreversible thermodynamics is shown. Equation for the entropy production in the interdiffusion process is displayed. The presented approach allows calculation of entropy production during interdiffusion, as well as formulating Onsagers phenomenological coefficients for the interdiffusion in an explicit form, a form which is directly correlated with the mobilities of the atoms present in the system.

Molecular Ion Channels; Electrodiffusion in R3

Ion transport across the membrane of the living cell (molecular ion channels) is a critical process, e.g., the triggering of nerve cells and heart muscle cells is coupled with mechanisms controlled by ion diffusion (electrodiffusion). Although the process is described by the century old Nernst- Planck-Poisson system of equations, it is not well understood and a clear understanding of how the interaction between channel and ions affects the flow is still missing. We present a three-dimensional model of the molecular channel. An appropriate quantitative description of the ion transport process allows proper explanation of molecule channel interactions (e.g. the ions flow for a given concentration gradient should depend on the potential and other parameters describing the interaction, i.e. asymmetric transport). We show the simulation of the stationary electrodiffusion in the ion channel showing radial symmetry.

Existence, uniqueness and properties of global weak solutions to interdiffusion with Vegard rule

We consider the diffusional transport in an

Viscosity Controlled Interdiffusion in Nitriding

r

Electrochemistry of Symmetrical Ion Channel: Three-Dimensional Nernst-Planck-Poisson Model

-component solid solution. The model is expressed by the nonlinear system of strongly coupled parabolic differential equations with initial and nonlinear boundary conditions. The techniques involved are the local mass conservation law for fluxes, which are a sum of the diffusional and Darken drift terms, and the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. The theorems on existence, uniqueness and properties of global weak solutions are proved. The main tool used in the proof of the existence result is the Galerkin approximation method. The agreement between the theoretical results, numerical simulations and experimental data is shown.

Diffusion Transport in Electrochemical Systems: A New Approach to Determining of the Membrane Potential at Steady State

This works presents new approach to model formation of expanded austenite (S-phase) during nitriding in plasma conditions. Diffusion saturation of the substrate (iron or austenite steel) is treated as interdiffusion of nitrogen and iron that involves stresses and plastic deformation and is based on the Darken scheme. It is argued that S-phase growing at nitriding behaves as elasto-viscous Maxwell solid. During the process, in the nitride zone, the dynamic pressure appears, which is related to Darken drift and depends on metal viscosity. Basic equations are formulated and discussed. The formula for drift is derived. Exemplary results, i.e. concentration profiles, dynamic pressure and dilatation of the sample during the process, are presented. Concentration profiles confirm existence of characteristic plateau like zone in the surface adjacent zone.