 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.   
 
