Transcript Lecture 15 Interfaces (cont.)

Module 20: Field–Assisted Diffusion
DIFFUSION IN SOLIDS
Professor Martin Eden Glicksman
Professor Afina Lupulescu
Rensselaer Polytechnic Institute
Troy, NY, 12180
USA
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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
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Background
The flux of a mobile component Ji related to the molar diffusion mobility tensor
{Bi} to a generalized force Fi
Ji  Ci Bi  Fi
The diffusion flux arising from the gradient of chemical potential,
Ji  Ci Bi i
The chemical potential of the ith component of a solution
defined relative to unit activity i  0  RgT lnai ,
Substituting
(0  ai  1)
J i   RgT Bi Ci
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Background
An additional flux term appears that arises from the electrostatic force acting
on the ions

Ci qi  
Ji   RgT Bi  Ci 

R
T


g
For an ideal or highly dilute solutions
Di  RgT Bi 
Substituting yields a form of Nernst’s equation:
 Ci qi 
Ji   Di Ci  Di 

R
T
 g 
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Diffusion currents in homogeneous ionic solids
The ion current flowing through a unit area of a homogeneous specimen is
Ci q2i 
Ji  Ji qi   Di 
 
 RgT 
The electronic current flowing in the detector circuit is given by
I ext
Ci q2i  A
 Di 
V
 

RgT  L 
Ohm’s law states that
I ext  Gi V
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Diffusion currents in homogeneous ionic solids
General expression for the electrical conductance in a
homogeneous conductor
Ci Di 2 A
Gi 
q i  
RgT L 
Ci Di
2 A
In terms of Faraday’s constant we may rewrite Gi 
ZF   
L 
RgT
Introducing the electrical conductivity, 
A
G   
L 
Substituting yields a basic formula for the
ionic conductivity
Ci Di
2

 ZF 
RgT
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Guard–ring circuit for ionic conductivity
measurements
+
Speci m en
Recor der
Amp.
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Measurements of ionic conductivity in solids
The mobile carriers in most ceramic materials
These transport numbers are defined so
I ext  J i e  a  c 

i
1
i
Defects in ionic solids
Schottky pairs
va + vc
Frenkel pairs
vc + ic
The most common ion–defect pairs
encountered are
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Equilibria Among Charged Point Defects
The concentration of Schottky pair defects
GSchottky 
Nva  Nvc  exp 

 kBT

The concentration of Frenkel pair defects
GFrenkel 
Ni c  Nvc  exp 
 kBT 

Overall
Nvc  Nva  Ni c
The tracer diffusivity of such an interstitial defect is Di c  D Ni
*
where
Ni 
Ci
C
The general conductivity equation

Z F 


Ci D
RgT
2
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Equilibria Among Charged Point Defects
Multiplying both sides by the concentrations
of the mobile ions
C
i
C Z F 

RgT
C D

2

 C Z F 
* 
Di c
RgT
2
Comparing
The interstitial tracer diffusivity Dic   
*
Considering that
RgT
C Z F 
2
D*va  f  D Nv

C Zv F 
* 
Dv
f RgT
2
The expression becomes
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Silicate Network with Mobile Cations
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Ionic Conductivity and Diffusivity in Soda–lime
Glass (J. Kelly’s experiments)
T (C)
D [cm2/sec]
 [ohm-cm]-1
f exp
200
250
300
4.2x10-13
4.0x10-12
2.6x10-11
5.05x10-8
3.25x10-7
1.51x10-6
0.23
0.29
0.38
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Irreversible Thermodynamics and Diffusion
1) Each transport flux depends linearly on all generalized thermodynamic forces
J1  L11 X1  L12 X 2  L13 X 3  ... L1n X n ,
J2  L21 X1  L22 X 2  L23 X 3  ... L2n X n ,
J3  L31 X1  L32 X 2  L33 X 3  ... L3n X n ,
Jn  Ln1 X1  Ln2 X 2  Ln3 X 3  ... Lnn X n .
2) The Onsager matrix of kinetic coefficients [Lik] is comprised of diagonal terms
[Lii]
Onsager’s reciprocity theorem
Lik = Lki
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Irreversible Thermodynamics and Diffusion
3) Each of the thermodynamic forces dissipates free energy and produces entropy
d Stot 
T 
   J i  Fi
 dT  i
List of the generalized forces associated with q, Ji, i
Flux, Ji
Force, Xi
Heat, q
1 
T  
T 
Diffusion, Ji
 
T  i 
 T 
Electron flow, i
E =-
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Isothermal Binary Diffusion
J1   L111  L12 2  L1v v ,
For the case of vacancy–assisted diffusion
J2   L211  L22 2  L2 v v ,
Jv   Lv11  Lv2  2  Lvv v .
The fluxes
J1  J2  Jv  0
L11  L21  Lv1 1  0,
Each term must vanish column by column
L12  L22  Lv2  2  0,
L1v  L2 v  Lvv  v  0.
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Isothermal Binary Diffusion
 Lv1  L11  L21 ,
The kinetic coefficient for the vacancy flux is dependent
on those for the component atoms
 Lv2  L12  L22 ,
 Lvv  L1v  L2v .
Combining with Onsager’s reciprocity relationship
J1   L111   v   L12 2   v ,
J2   L121   v  L22 2   v .
The vacancy flux can be written
Jv  L1v1  v  L2v2   v 
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Vacancies in Thermal Equilibrium
The chemical potential for a binary alloy is
J A  LAA A  LAB B ,
J B  LAB A   LBB B .
 A   0A  kBT lnCA,
If in the binary alloy the lattice vacancies
B



B
0  kBT lnCB ,
form an ideal solution
 v  0.
Writing in terms of the component concentrations
J A  LAAkBT lnCA   LABkBT lnCB ,
J B  LABkBT lnCA  LBBkBT lnCB .
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Vacancies in Thermal Equilibrium
or
LAA 
LAB 
J A    kBT CA   kBT  CB,
CA 
CB 
LAB 
LBB 
J B    kBT  CA   kBT  CB.
CA 
CB 
Simplifying
LAA 
J A    kBT CA,
CA 
LBB 
J B    kBT CB .
CB 
The equations have a similar form with Fick’s
first law
Ji  Di Ci , i = A, B
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Vacancies in Thermal Equilibrium
The intrinsic diffusivities are
 Lii 
Di  kBT  
Ci 
 L AA  
D A  kBT 
, 

 C A  

The intrinsic diffusivities can be expressed as

 LBB  
DB  kBT 
, 

 CB  

D A  DB .
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Net Vacancy Flux
The Gibbs–Duhem relationship provides that
CA A  CB B
LAA LAB 

J A    
kBT CA,


CA CB 

It then follows that
LBB LAB 

J B    
kBT CB.


CB CA 

 LAA LAB  
DA  kBT 

,

 CA CB  

For non–reciprocal binary diffusion,

Onsager’s equations show that
 LBB LAB  
DB  kBT 

,

CA  
 CB

DA  DB.
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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?
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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

 
Je  Le e    Le C  C ,
x
 x 

 C 
JC   LCe   LCC 
.

x 
 x 
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
Lee
Je 

 J 
 C C0 LCe
Je 
LeC



JC 0 LCC
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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 nonreciprocal 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.
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