Embed Size (px)
Citation preview
© meg/aol ‘02
Module 20: Field–Assisted Diffusion
DIFFUSION IN SOLIDS
Professor Martin Eden Glicksman
Professor Afina Lupulescu
Rensselaer Polytechnic Institute
Troy, NY, 12180
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
Ji Ci Bi Fi
Ji Ci Bi i
i 0 RgT lnai , (0 ai 1)
Ji RgT Bi Ci
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
© meg/aol ‘02
Background
Ji RgT Bi Ci CiqiRgT
Di RgT Bi
Ji DiCi Di
Ci qiRgT
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
Ji Jiqi Di
Ci qi
2
RgT
Iext Di
Ciqi
2
RgT
ALV
Iext Gi V
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
Gi CiDi
RgTq
i
2 AL
Gi Ci Di
RgTZF 2 A
L
GAL
CiDi
RgTZF 2
General expression for the electrical conductance in ahomogeneous 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
Recor der
Amp.
Specimen+
© meg/aol ‘02
Measurements of ionic conductivity in solids
Iext J i e a c
ii 1
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 va + vc
Frenkel pairs vc + ic
© meg/aol ‘02
Equilibria Among Charged Point Defects
NvaNvc
exp GSchottky
kBT
The concentration of Schottky pair defects
The concentration of Frenkel pair defects
NicNvc
exp GFrenkel
kBT
NvcNva
NicOverall
Dic* DNi
Ni
Ci
C
Ci D
ZF 2RgT
The tracer diffusivity of such an interstitial defect is
where
The general conductivity equation
© meg/aol ‘02
Equilibria Among Charged Point Defects
Ci
C D
C ZF 2
RgT
D
i c
* C ZF 2
RgT
Dic
* RgT
C ZF 2
Multiplying both sides by the concentrations of the mobile ions
Comparing
The interstitial tracer diffusivity
Considering that
Dva
* fDNv
The expression becomes
D
v
* C ZvF 2
fRgT
© meg/aol ‘02
Silicate Network with Mobile Cations
Na+ O2-Si4+
© meg/aol ‘02
Ionic Conductivity and Diffusivity in Soda–lime Glass (J. Kelly’s experiments)
T (C) D [cm2/sec] [ohm-cm]-1 f exp
200 4.2x10-13 5.05x10-8 0.23250 4.0x10-12 3.25x10-7 0.29300 2.6x10-11 1.51x10-6 0.38
© meg/aol ‘02
Irreversible Thermodynamics and Diffusion
J 1 L11X1 L12 X2 L13 X3 ... L1nXn,
J 2 L21X1 L22 X2 L23 X3 ... L2nXn,
J 3 L31 X1 L32 X2 L33 X3 ... L3nXn,
J n Ln1 X1 Ln2 X2 Ln3 X3 ... LnnXn.
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
© meg/aol ‘02
Irreversible Thermodynamics and Diffusion
TdStot
dT
Ji Fi
i
3) Each of the thermodynamic forces dissipates free energy and produces entropy
Flux, Ji Heat, q Diffusion, Ji
Electron flow, i
Force, Xi T1
T T
i
T E=-
List of the generalized forces associated with q, Ji, i
© meg/aol ‘02
Isothermal Binary Diffusion
J 1 L111 L122 L1vv,
J 2 L211 L222 L2 vv,
J v Lv11 Lv22 Lvvv.
J 1 J 2 J v0
L11 L21 Lv1 1 0,
L12 L22 Lv2 2 0,
L1v L2 v Lvv v 0.
For the case of vacancy–assisted diffusion
The fluxes
Each term must vanish column by column
© meg/aol ‘02
Isothermal Binary Diffusion
Lv1 L11 L21,
Lv2 L12 L22 ,
Lvv L1v L2v.
J 1 L11 1 v L12 2 v ,
J 2 L12 1 v L22 2 v .
J v L1v 1 v L2v 2 v
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
J A LAA A LAB B ,
J B LAB A LBB B .
A 0A kBT lnCA,
B 0B kBT lnCB,
v 0.
J A LAAkBT lnCA LABkBT lnCB ,
J B LABkBT lnCA LBBkBT lnCB .
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’sfirst law
J A LAA
CA
kBT CA
LAB
CB
kBT CB,
J B LAB
CA
kBT CA
LBB
CB
kBT CB.
J A LAA
CA
kBT CA,
J B LBB
CB
kBT CB.
J i Di Ci , i =A,B
Vacancies in Thermal Equilibrium
or
Simplifying
© meg/aol ‘02
Di kBTLii
Ci
DAkBTLAA
CA
,
DB kBTLBB
CB
,
DADB.
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
CAA CBB
It then follows that
For non–reciprocal binary diffusion, Onsager’s equations show that
DAkBTLAA
CA
LAB
CB
,
DB kBTLBB
CB
LAB
CA
,
DA DB.
J A LAA
CA
LAB
CB
kBT
CA,
J B LBB
CB
LAB
CA
kBT
CB.
© 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
Je L
e ex L
e C
C
x
,
J C LCe
x LCC
C
x
.
Je
J C
C0
L
e e
LCe
J e
J C
0
LeC
LCC
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 (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.
