Page 632 - Handbook of Battery Materials

606 17 Liquid Nonaqueous Electrolytes

Actually developed for polymer electrolytes [467], the method is also used for liquid
electrolytes. Nonetheless, the drawback is the time-consuming procedure until
equilibrium is adjusted.
Every method has its advantages and disadvantages, and it is remarkable that
different methods give different results. There is always a model applied on the
calculation and it should be acted with caution which method is applicable. As can
be seen in Table 17.18, measured transference numbers vary widely with different
methods, caused by the different assumptions made in every method. In Ref.
[456], two different methods were used for the same system and in accordance
with investigations from this group [528], completely contrary tendencies of the
concentration dependence of the transference number were obtained. Nevertheless,
the transference number is an important parameter, which is also reflected by
increasing interest in recent years.

17.4.7
Diffusion Coefficients in Liquids

17.4.7.1 Introduction
For an adequate description of transport phenomena the diffusion coefficient
should not be left out. The performance of lithium-ion batteries is directly connected
to the mass transport in the electrolyte, as studied by Sawai et al. [479]. He showed
that the lithium-ion diffusion coefficient in the solution is even more important for
the rate capability of graphite than the diffusion of lithium in the solid electrode.
Although being one of the key determinants for the power output, very few
diffusion data in liquid electrolytes are available. This has not changed since 1947,
when Harned wrote: ‘There are few domains of physical science in which so much
experimental effort over nearly a century has yielded so little accurate data as the
field of diffusion in liquid systems’ [480].
For highly diluted systems an exact theory is modeled [481], but concentrated
solutions bring complex problems, such as short-range forces or convection.
Generally, diffusion can be described by Fick’s first law [472]:
∂c i
J i =−A·D (17.71)
∂x
with J i as the total one-dimensional flux, A the area across which diffusion occurs,
D the diffusion coefficient, c i the concentration, and x the distance. In the original
paper the diffusion coefficient is indicated as ‘constant dependent upon the nature
of the substances’ [472].
To estimate diffusion coefficients in liquid electrolytes, the Stokes-Einstein-
relation [482] is generally used:
k B T
D = (17.72)
6πηr
with k B Boltzmann’s constant, T temperature, η solvent viscosity, and r solute
radius. This equation is limited to systems with a solute radius five times larger than
the solvent radius, and a simple model is assumed in which a rigid sphere moves