Page 157 - Lindens Handbook of Batteries

6.10 PRINCIPLES OF OPERATION

6.4.2 Charge Transport by Ions
A unique feature of electrochemical devices is the transport of the current by ions. Once the current
moves past the electrodes and undergoes the electrochemical reaction, the charge transport from one
electrode to another is facilitated by ions. This transport of charge by the movement of ions is more
complicated than the current carrying mechanism by movement of electrons. Usually, there are sev-
eral species of ions present in the electrolyte. The total current density carried by the electrolyte (i )
2
across a unit normal area is the sum of the current density carried by each species k
i = ∑ i (6.14)
k
2
k
The current density carried by species k, i is proportional to its flux 7
k
i = ∑ N (6.15)
F
k
k
k
The proportionality factor in Eq. (6.15) is the Faraday’s constant, which is the amount of charge car-
ried by each mole of the ion. The flux of ion k may be defined as the product of the number of k ions
per unit volume of the electrolyte (i.e., the concentration of species k) and the velocity of each ion
N = c v (6.16)
k k
k
The concentration of the electrolyte is a readily measurable quantity; the velocity of an ion is pro-
portional to the charge it carries (z ) and the potential gradient in the solution phase (∇φ 2 ) which is
k
the electrical driving force for that ion to move
v = -u Fz ∇φ (6.17)
2
k
k
k
The proportionality constant in Eq. (6.17) is called the mobility (u ) of the ion and is obtained from
k
equivalent conductance measurements. The negative sign indicates that the ions move from a region
of higher potential to lower. Equations (6.14) to (6.17) can be rearranged to obtain. 7
 
2
i =  F ∑ cu z ∇φ (6.18)
kk k 
2
 k  2
Equation (6.18) resembles Ohm’s law closely, and the electrical conductivity for the electrolyte (κ)
is now given by
 
κ= F 2  (6.19)
 ∑ c uz
 k kk k 
As stated in the previous section (see Eq. [6.13]), Eq. (6.19) relates the properties of the component
ions to the conductivity of the electrolyte. Hence knowing the composition of the electrolyte, one
can model the movement of ions in the electrolyte. Alternatively, one can experimentally measure
the electrical conductivity (κ) in Eq. (6.19).
In deriving Eq. (6.18), an implicit assumption that the concentration of the electrolyte was uniform
precluded the effects of concentration gradients present within the cell. However, this assumption
can be easily relaxed by incorporating flux terms arising from concentration differences using Fick’s
laws of diffusion. Equation (6.16) then becomes

c
N = k c v - k k D ∇ (6.20)
k
k