Joshua Flygare and Dr. Dean Wheeler, Chemical Engineering
Introduction: There are many significant barriers still to be overcome before chemical storage becomes satisfactory. One of the most significant issues is how much the electrodes vary in their electronic and ionic conductivities. Both of these issues are highly complex. My funding was used to help me learn the basic mathematics and physics that occur inside of batteries. I used this understanding to make two models, one using COMSOL multiphysics, and another cruder model using Excel. The most significant change made in my execution from my original proposal was I used all of my time in understanding and making models, and did not reach a point of doing measurements.
Methodology: The fundamental equations to be solved for entire electrochemical systems are extremely complex; particularly the movement of ions in the electrolyte. This is difficult to model because there are so many factors that influence the diffusion and convection occurring in a non-ideal liquid. The equations for electronic conductivity and ionic movement must be solved simultaneously because they are coupled, meaning they effect one another. These equations do not have an analytical solution unless they are simplified greatly, especially when applying boundary conditions. They must be solved numerically using advanced numerical methods. A large portion of my time was spent reviewing vector calculus and learning the physics that occurs in the battery. A full scale model written entirely alone is a PhD level endeavor.
Results/Discussion: The simple model made in excel was to showcase how a multi-dimensional elliptical differential equation (The Laplace equation) can be crudely approximated. Figure 1 shows this approximation and highlights the elliptical nature of the solution. This solution is only for the electronic conductivity of a thin-film cathode attached to a metal current collector. The Laplace equation only shows how the voltage distribution changes due to the material resistance. A much more complex model was made using COMSOL multiphysics by a graduate student and Dr. Wheeler. I extended this model to be multi-dimensional. This model solves the solutions simultaneously for both the ionic and electronic conductivities. This solution is obtained by breaking up the equations into a “mesh” that cover the geometry of the system. Then, inside of each of those sections that make up the mesh, the differential equations are solved over a fixed change in distance, forcing them to become only algebraic expressions. Figure 2 shows the results of this model.
Conclusion: This grant provided me an excellent opportunity to get a head start on my eventual dissertation that will require advanced modeling techniques. I learned a great deal about the physics of batteries and the math necessary to apply that physics. The models I made are by no means finished products, but they do serve as stepping stones to make more accurate and demanding models.