Heat of Mixing in Intercalating Electrodes
For intercalating porous electrodes in the battery interfaces, the heat of mixing
q
mix,
i
(W/m
3
) of a species of index
i
is defined as (see
Ref. 4
)
where
N
i
(mol/(m
2
s)) is the molar flux and
i
(J/mol) is the partial molar enthalpy.
This expression could be seen as a species transport version of Joule heating in an electronic conductor, for which the heat source is defined as the current density multiplied by the gradient of the potential.
Generally, for an intercalation material, two species are considered: the intercalated species, denoted with index
s
, and the holes, denoted with index
θ
.
The flux of the two species are equal and opposite.
We now define the total heat of mixing as the sum of the contributions from the two species and write
The absolute value of the individual molar enthalpies are generally not known. However, the difference of the gradients of the molar enthalpies are related to the thermoneutral voltage,
E
therm
(V) of the intercalation reaction.
The thermoneutral voltage is defined as
where
Δ
H
is the change of molar enthalpy of the redox reaction.
In the case of lithium intercalation the redox reaction is
with index
s
= Li(s) so that
We now note that, if assuming the equilibrium potential to be independent of the electrolyte lithium ion concentration,
and finally we get
i
