
Electrical current can conducted by
ions in
 Liquid electrolytes (like H_{2}SO_{4} in your
"lead  acid" car battery); including gels
 Solid electrolytes (= ionconducting crystals). Mandatory for fuel cells
and sensors
 Ion beams. Used in (expensive) machinery for "nanoprocessing".


Challenge: Find / design a material with a
"good" ion conductivity at room temperature 


Basic principle 




Diffusion
current j_{diff} driven by concentration
gradients grad(c) of the charged particles (= ions here)
equilibrates with the 

j_{field} = s
· E = q · c
· µ · E 




Field current
j_{field} caused by the internal field always associated
to concentration gradients of charged particles plus the field coming from the
outside 



Diffusion coefficient D and
mobility µ are linked via theEinstein relation;
concentration c(x) and potential U(x)
or field E(x) =
–dU/dxby the Poisson equation. 

– 
d^{2}U
dx^{2} 
= 
dE
dx

= 
e ·
c(x)_{ }
ee_{0}









Immediate results of the equations
from above are: 




In equilibrium we find a preserved quantity, i.e.
a quantity independent of x  the electrochemical potential
V_{ec}: 

V_{ec} 
= const. = 
e · U(x) + 
kT 
· ln c(x) 




If you rewrite the equaiton for
c(x), it simply asserts that the particles are distributed
on the energy scale according to the Boltzmann distrubution: 

c(x) = exp – 
(Vx) – V_{ec}
kT 




Electrical field gradients and concentration gradients at "contacts" are coupled and
nonzero on a length scale given by the Debye length
d_{Debye} 

d_{Debye} = 
æ
ç
è 
e ·
e_{0} · kT
e^{2} · c_{0} 
ö
÷
ø 
1/2 




The Debye length is an extremely important
material parameter in "ionics" (akin to
the space charge region width in semiconductors); it depends on temperature
T and in particular on the (bulk) concentration
c_{0} of the (ionic) carriers. 



The Debye length is not an important material
parameter in metals since it is so small that it doesn't matter much. 







The potential difference between two
materials (her ionic conductors) in close contact thus... 




... extends over a length given (approximately)
by : 

d_{Debye}(1) +
d_{Debye}(2) 




... is directly given by the Boltzmann
distribution written for the energy:
(with the c_{i} =equilibrium conc. far away from the
contact. 

c_{1}
c_{2} 
= exp – 
e · DU
kT 

Boltz
mann 
DU = –

kT
e 
· ln 
c_{1}
c_{2} 

Nernst's
equation 




The famous Nernst
equation, fundamental to ionics, is thus just the Boltzmann
distribution in disguise! 


"Ionic" sensors (most
famous the ZrO_{2}  based O_{2} sensor in your
car exhaust system) produce a voltage according to the Nernst equation because
the concentration of ions on the exposed side depends somehow on the
concentration of the species to be measured. 




