Module 20: Field–Assisted Diffusion

Module 20: Field–Assisted Diffusion DIFFUSION IN SOLIDS Professor Martin Eden Glicksman Professor Afina Lupulescu Rensselaer Polytechnic Institute Troy, NY, 12180 USA

Outline • Background • Diffusion currents in homogeneous ionic solids • Measurement of ionic conductivity in solids • Defects in ionic solids • Experiments in ionic conductors • Irreversible thermodynamics and diffusion • Isothermal binary diffusion • Vacancies in thermal equilibrium • Net vacancy flux • Exercise

Background The flux of a mobile component Ji related to the molar diffusion mobility tensor {Bi} to a generalized force Fi The diffusion flux arising from the gradient of chemical potential, The chemical potential of the ith component of a solution defined relative to unit activity Substituting

Background An additional flux term appears that arises from the electrostatic force acting on the ions For an ideal or highly dilute solutions Substituting yields a form of Nernst’s equation:

Diffusion currents in homogeneous ionic solids The ion current flowing through a unit area of a homogeneous specimen is The electronic current flowing in the detector circuit is given by Ohm’s law states that

Diffusion currents in homogeneous ionic solids General expression for the electrical conductance in a homogeneous conductor In terms of Faraday’s constant we may rewrite Introducing the electrical conductivity,  Substituting yields a basic formula for the ionic conductivity

Guard–ring circuit for ionic conductivity measurements

Measurements of ionic conductivity in solids The mobile carriers in most ceramic materials These transport numbers are defined so Defects in ionic solids Schottky pairsva + vc The most common ion–defect pairs encountered are Frenkel pairsvc + ic

The concentration of Schottky pair defects The concentration of Frenkel pair defects Overall The tracer diffusivity of such an interstitial defect is where The general conductivity equation Equilibria Among Charged Point Defects

Equilibria Among Charged Point Defects Multiplying both sides by the concentrations of the mobile ions Comparing The interstitial tracer diffusivity Considering that The expression becomes

Silicate Network with Mobile Cations

Ionic Conductivity and Diffusivity in Soda–lime Glass (J. Kelly’s experiments)

Irreversible Thermodynamics and Diffusion 1) Each transport flux depends linearly on all generalized thermodynamic forces 2) The Onsager matrix of kinetic coefficients [Lik] is comprised of diagonal terms [Lii] Onsager’s reciprocity theorem Lik = Lki

List of the generalized forces associated with q, Ji, i Irreversible Thermodynamics and Diffusion 3) Each of the thermodynamic forces dissipates free energy and produces entropy

Isothermal Binary Diffusion For the case of vacancy–assisted diffusion The fluxes Each term must vanish column by column

Isothermal Binary Diffusion The kinetic coefficient for the vacancy flux is dependent on those for the component atoms Combining with Onsager’s reciprocity relationship The vacancy flux can be written

Vacancies in Thermal Equilibrium The chemical potential for a binary alloy is If in the binary alloy the lattice vacancies form an ideal solution Writing in terms of the component concentrations

Vacancies in Thermal Equilibrium or Simplifying The equations have a similar form with Fick’s first law

Vacancies in Thermal Equilibrium The intrinsic diffusivities are The intrinsic diffusivities can be expressed as

Net Vacancy Flux The Gibbs–Duhem relationship provides that It then follows that For non–reciprocal binary diffusion, Onsager’s equations show that

Exercise 1. When a DC voltage is applied to a bar of –Fe containing a homogeneous interstitial solution of carbon, an electric current flows and carbon atoms migrate and collect near the cathode. This “unmixing” process is called electro–diffusion, and is analogous to the ionic diffusion in a homogeneous ceramic solids described in §20.2. a) Develop the linear phenomenological equations for electro–diffusion, assuming that the iron atoms, at the temperature of this experiment, are immobile compared to the highly mobile carbon interstitials, and that the current measured is comprised virtually entirely of electrons. b) Find the ratio of the electron flux to the carbon flux in a homogeneous Fe–C specimen. c) If the applied voltage is zero, show that a charge still flows when the carbon atoms diffuse under their own gradient of chemical potential. What is the nature of this charge?

Solution In analogy to the ionic diffusion case, the flux of each mobile quantity depends on all the thermodynamic forces—in this example the Coulomb force and the generalized chemical force. For electrons (e-) and carbon (C) one can write individual flux equations B) The ratio of the electron flux to the carbon flux when the chemical potential gradient vanishes (a homogeneous alloy) is given by the ratio of the expressions shown in eq.(20.47) and setting (/x)=0. C) The ratio of the electron flux to the carbon flux when the voltage is absent is given by the ratio of the expressions shown in eq.(20.47), and setting (/x)=0; hence

Key Points • In ionic conductors, anions and cations diffusion on their separate lattices. • To preserve electrical neutrality, special defects, in addition to lattice vacancies are needed: • Frenkel pairs (vc+ ic) • Shottkey pairs (vc + va) • Atom fluxes are generally composed of a diffusive term and a drift term, caused by an external field. • In ionic conductors, the flux can be driven by an applied potential, and the net charge flux senses as an external current. • Tracer studies in soda-lime glasses discussed. • Onsager’s irreversible thermodynamic formalism applied to diffusion in solids. Intrinsic diffusivities for reciprocal (no net vacancies) and non-reciprocal binary diffusion (vacancy wind) may be expressed in terms of the Onsager kinetic coefficients. • Electro-diffusion (akin to electrolysis) can be formulated similarly, to show that the flow of electrons moves atoms, and the flow of atoms moves electrons.

Module 20: Field–Assisted Diffusion