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Theory of Charge Transport in Mixed Conductors: Description of Interfacial Contributions Compatible with the Gibbs Thermodynamics
Mikhail A. Vorotyntsev
LSEO-UMR 5188 CNRS, UniversitИ de Bourgogne, Dijon, France


Content
· Introduction: mixed transport. · Randles impedance. Interfacial capacitance. Formulation of the problem. · Thermodynamics of charged interfaces. Conditions at interfaces for transport. · Analytical expressions for impedance. Graphical illustrations. · New systems: "mixed interfacial exchange". · Conclusions.


medium 1 film : 2 mobile species plus fixed charges medium 2

Film ( f ) : - conducting or redox polymers, - electron-ion conducting oxides/hydrids/sulfides, + ++ - Li & Mg intercalation layers, - solid electrolytes, -thin layers of a binary solution.


Species : - electronic and ionic (e, i) plus fixed charge, - cations and anions.

Media 1 et 2 : - electronic conductor ( m ), - ionic conductor ( s ).
Tree types of systems : - m/f/m' : between two electronic conductors, - s'/f/s : "membrane geometry", - m/f/s : "modified electrode"


Electrochemical Impedance Spectroscopy - transport properties, - interfacial characteristics Based on : - analytical analysis, - equivalent circuit. To be included : - bulk film transport, - charge transfer across the interfaces ("faradaic"), - charge of interfaces (charge of "double layers")


Randles Impedance : metal / solution Supporting electrolyte + Redox species

C Rs W
General hypothesis :

dl

R

ct

- consider the process without interfacial charge, - add "double layer capacitance" C parallel to the "faradaic branch".
dl


System containing only two mobile species: - no supporting electrolyte, - the same species participate at each interface - in the "faradaic" process (redox reaction or ion exchange) as well as - in the interfacial charge

these two processes are coupled to the same transport process (Warburg element).


Another complication :
metal EDL + , diffusion layer bulk solution ik = tk i

-

Composition of the double layer (charges + , -) is determined by properties of the interface.
Partial currents ik in the bulk solution are determined by transport numbers tk .


This discrepancy of the partial currents and the variation of charges + - must be compensated by the diffusion layer
,

its impedance (analogue of W) must be a function of tk as well as of parameters determining + and -

Conclusion : capacitance C is not sufficient to characterize the charging of interfaces !

dl


Thermodynamics of interfaces ­ binary solution: 2 independent variables, e.g. & = +/z+ - -/zsolution: dE+ = (Cdl)-1 d - t -dl d(/F) d+ = - t+dl d - Cdl d(/F) EDL: + , d- = - t -dl d + Cdl d(/F) metal: THREE independent interfacial parameters: (1) Cdl , interfacial capacitance ("capacitance of the electrical double layer"), (2) t+dl et t-dl , "interfacial numbers of species" : dl dl + + - = - t+ + t - = 1 The same coefficient t -dl for dE+ et for d- ,

(3) C , "asymmetry factor of the interfacial charge"

dl


E le c tr olyte w ith out s p e c ific ads o r p tion th e o ry o f G o u y -C h a p m a n -G ra h a m e - th e s e p a ra m e te rs a re fu n c tio n s o f and - ge ne ra lly, C dl ~ C dl ; 0 t+ dl, t -dl S p e c ific a d so rp tio n : tkdl < 0 or tkdl >

: , 1 1

ik(0,t) = ikdl + ikct ikdl = - dk / dt ct ext ik = [ k - k(0,t) ] / zkF R

k

ct

( Ek ­ Ek°) / R

k

ct

Transport equations + conditions at interfaces
Their combined solution gives analytical expressions of impedance Z() for 3 geometries of the system :

m/f/m' (different metals), m/f/s, s'/f/s.


m/f/m : film between two identical metals Z Z
m/f/m m/f

= Rf + 2 Z
m/f Re -1

m/f

+ 4 Wf ( ti ) [coth + F ]
m/f -1

2

m/f -1

=(1/

+ j C

)

Wf = Rf ; = ( j L / 4D ) ti = ti ­ tim/f (1 - gm/f ) ; gm/f = (1 + j Rem/f C m/f m/f m/f 2 m/f m/f F = 2 Wf j [ ( ti ) g C - C ]

2

1/2

m/f -1

)


m/f/s : Z
m/f/s

film between a metal and a solution
m/f

= Rs + Rf + Z
2

+ Z + 2 W f Za / Z
f/s

f/s

b -1

Za = ( ti ) ( coth 2 + F ) + 2 ti te ( sinh 2 ) 2 m/f + ( te ) ( coth 2 + F ) ; m/f f/s m/f f/s Zb = 1 + ( F + F ) coth 2 + F F


s/f/s : Z
s/f/s f/s

particular case,
f/s

f/s te

= 0, C



f/s

=0

= Rf + 2 Z + 4
f/s Ri -1

2 Wf te f/s -1

tanh

Z =(1/

+ j C )

Wf = Rf ;

= ( j L / 4D )

2

1/2


Symmetrical membrane geometry: film between two identical solutions, s/f/s
100

1: 2: 3: C

Cf/s Cf/s Cf/s f/s =

=1 = 30 = 1000 0, tef/s = 0

- Im Z

50

3 2 1

0

0

50

100

150

200

Re Z


100

1: 2: 3: C

Cf/s Cf/s Cf/s f/s =

=1 = 30 = 1000 0, tef/s = 0

- Im Z

50

3 2 1

0 100 0

50

100

150

200

1: tef/s = 0 2: tef/s = 0,5 3: tef/s = 1

Re Z

- Im Z

50

Cf/s = 30, C f/s = 0

2 3
0 0 50

1

100

150

200

Re Z


100

0: C f/s = 0 1: C f/s = 1 2: C f/s = 10 3: C f/s = 30

tef/s = 0, Cf/s = 30

- Im Z

50

2 1
0 0

1 3

2 3
50

100

150

200

Re Z


Modified Electrode Geometry: film between a metal and a solution, m/f/s
50

1: C mf=30, C fs=1 2: C mf=30, C fs=30 3: C mf=30, C fs=1000

temf=1, C tifs=1,



mf

=0

40

Cfs=0 3 2 1

30

- Im Z

20

10

0

0

20

40

60

80

100

Re Z


50

1: C mf=30, C fs=1 2: C mf=30, C fs=30 3: C mf=30, C fs=1000 temf=0, C ti =0,
fs


40

3 2 1

30

mf fs

=0

- Im Z

20

C =0

10

0

0

20

40

60

80

100

Re Z


metal (e) electron exchange film : e + i ion exchange solution (i)

metal (e) electron exchange film : e + i electron + ion exchange electroactive solution (e,i): Red Ox + ne


New geometries: m/f/es, es'/f/es, s/f/es
Similar treatment of boundary conditions for transport

Analytical solutions for all new geometries
New experimental possibilities: one can obtain impedance data for numerous systems having the same values of the bulk film and interfacial parameters


CONCLUSIONS

- there is no simple way to insert the contribution of the interfacial charge in the final expression for complex impedance.
- contrary to expectations, interfacial capacitance Cdl is not sufficient to characterize this contribution : impedance also depends on interfacial numbers t±dl as well as parameter Cdl.


- thin film with a mixed conductivity : one can obtain analytical expressions Z() in the cases : 1. between two metals (identical or different), 2. between two solutions ("membrane geometry"), 3. between a metal and a solution ("modified electrode").


- if the charging of the double layer is realized completely by the "faradaic" species, the effect of the interfacial charge is very simple : capacitance Cdl in parallel to Rem/f or Rif/s.

- general case : impedance plots are markedly deformed with respect to this simple case.
Application of simplified formulae can lead to serious errors in the value of capacitance Cdl found from the treatment of experimental data.


- new prospects to extract the bulk-film and interfacial parameters of the system are provided by "non-traditional" arrangements, films in contact with "electroactive solutions".

Analytical formulae for complex impedance are now available for all possible 1D geometries:

m'/f/m, s'f/s, m/f/s, m/f/es, es'/f/es, s/f/es