WPI Mathematical Problems in Industry

MPI Home
About MPI

© 2008 CIMS

Modeling Battery Electrode Properties

Jacqueline Ashmore & David Clatterbuck, TIAX LLC

Many (electro)chemical systems contain active components (i.e., electrodes, catalysts, or reactors) which are made from packed particles. Some examples include anodes and cathodes in both Li-ion batteries and solid-oxide fuel cells, and fluidized bed reactors. Often important reactions take place on the surface of the particles while the necessary mass and/or electrical transport occurs through the void space between particles (which may be filled with a fluid), through the individual particles themselves, and through the percolating network of interconnected particles. In many cases, the amount of solids in a given volume (packing fraction) is also important. Thus it would be helpful to have methods for determining how a set of physical characteristics of the component particles (i.e., particle size, particle shape, surface roughness) affect the properties of the compacted porous body (i.e., total surface area, void fraction, degree of interconnectedness).

A brief description of some of physical processes that occur in the battery is as follows: during charging of the battery, a Li atom starts inside a solid particle of the cathode material. It diffuses through the solid to the surface of the particle and disassociates from the particle, creating a Li ion. This ion travels through a liquid electrolyte that occupies the void volume of the electrode. It is then transported through the electrolyte to the other electrode (anode) made of a different material where it again reacts at the surface of a solid particle, enters a solid particle, and finally undergoes solid state diffusion within the anode particle. Below are two typical problems encountered in the design of Li-ion batteries where the cathode consists of a porous network of packed particles of the active material.

Example 1. For a given battery chemistry, the total energy content of the battery increases with the amount of active material and is thus proportional to the packing fraction for a battery of fixed volume. At the same time, the rate of energy delivery (i.e., power) depends on having sufficient mass and electrical transport throughout the electrode. Thus, a battery may be designed to give a high volumetric energy density by increasing the packing fraction, while in another application where high power is needed, a more porous electrode may be used.

Example 2. For some materials, Li transport through the bulk of the individual particles will be the rate limiting step, and thus it is desirable to decrease the particle size to allow for shorter transport distances (diffusion lengths). However, it has been found empirically that very small particles cannot be packed as densely as coarse particles due to surface effects; thus a practical electrode will likely have a particle size which is a compromise between these two effects.

Statement of problem
The geometry of a porous electrode made of packed particles plays an important role in determining the performance of many electrochemical systems. TIAX would benefit from algorithms, methods, models, scaling relations, or frameworks to analyze the effect of different particle characteristics on electrode properties.

Some particle characteristics to consider might include:

  • size of monodisperse spheres
  • roughness of monodisperse spheres
  • radii of bidisperse spheres
  • particle sizes with more realistic distributions of sizes (i.e. Gaussian distribution)
  • deviations from sphericity, e.g., ellipsoidal particles

Some electrode properties of interest include:

  • Packing fraction or void volume
  • Total surface area
  • Average path lengths for transport through the individual solid particles to the particle surface
  • Average path length for diffusion through the void volume from the surface of a particle to the surface of a collection of particles (electrode) of a certain thickness. Effective cross-sectional area for this type of mass transport.
  • Average path length to travel through the collection of particles of a certain thickness, if you must travel only through the particles (passing from particle to particle only at points where they meet). Effective cross-sectional area for this type of transport.

Determining all of the electrode properties for all possible combinations of particle characteristics is probably not a manageable task. Rather it may be useful to consider some of the more complex electrode properties for the case of simple particle size distributions (i.e., monodisperse spheres); while for more complex particles, determining the packing fraction may be a sufficiently challenging problem.

WPI [Math] Back Top

Last Changed: Tuesday, 03-Jun-2008 10:11:37 EDT