© meg/aol ‘02 Module 20: Field–Assisted Diffusion DIFFUSION IN SOLIDS Professor Martin Eden Glicksman Professor Afina Lupulescu Rensselaer Polytechnic Institute Troy, NY, USA

© meg/aol ‘02 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

© meg/aol ‘02 Background The flux of a mobile component J i related to the molar diffusion mobility tensor {B i } to a generalized force F i The diffusion flux arising from the gradient of chemical potential, The chemical potential of the i th component of a solution defined relative to unit activity Substituting

© meg/aol ‘02 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:

© meg/aol ‘02 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

© meg/aol ‘02 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

© meg/aol ‘02 Guard–ring circuit for ionic conductivity measurements

© meg/aol ‘02 Measurements of ionic conductivity in solids The mobile carriers in most ceramic materials These transport numbers are defined so Defects in ionic solids The most common ion–defect pairs encountered are Schottky pairs v a + v c Frenkel pairs v c + i c

© meg/aol ‘02 Equilibria Among Charged Point Defects 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

© meg/aol ‘02 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

© meg/aol ‘02 Silicate Network with Mobile Cations

© meg/aol ‘02 Ionic Conductivity and Diffusivity in Soda–lime Glass (J. Kelly’s experiments)

© meg/aol ‘02 Irreversible Thermodynamics and Diffusion 1) Each transport flux depends linearly on all generalized thermodynamic forces 2) The Onsager matrix of kinetic coefficients [L ik ] is comprised of diagonal terms [L ii ] Onsager’s reciprocity theorem L ik = L ki

© meg/aol ‘02 Irreversible Thermodynamics and Diffusion 3) Each of the thermodynamic forces dissipates free energy and produces entropy List of the generalized forces associated with q, J i, i

© meg/aol ‘02 Isothermal Binary Diffusion For the case of vacancy–assisted diffusion The fluxes Each term must vanish column by column

© meg/aol ‘02 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

© meg/aol ‘02 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

© meg/aol ‘02 The equations have a similar form with Fick’s first law Vacancies in Thermal Equilibrium or Simplifying

© meg/aol ‘02 Vacancies in Thermal Equilibrium The intrinsic diffusivities are The intrinsic diffusivities can be expressed as

© meg/aol ‘02 Net Vacancy Flux The Gibbs–Duhem relationship provides that It then follows that For non–reciprocal binary diffusion, Onsager’s equations show that

© meg/aol ‘02 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?

© meg/aol ‘02 Solution A)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

© meg/aol ‘02 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 (v c + i c ) –Shottkey pairs (v c + v a ) 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.