MATSE 419 | Computational materials science and engineering

Laboratory 2 | BATTERY CHARGING & DISCHARGING

Due date

Submit your work by midnight of the due date on CANVAS. There are several options for submission: (i) a written report as PDF together with your code as .ipynb or .m, (ii) a single .ipynb notebook with markdown-formatted answers to the questions below (e.g., using “markdown” and equation formatting for your derivations, NOT Python comments).

Line numbers are indicated for the .m MATLAB codes here. Equivalent line numbers are indicated directly in the .ipynb notebook for Python codes. The line numbers are not the same between MATLAB and Python.

Objective

The objective of the study is to model and simulate the sorption/desorption of lithium ions in the electrode of a lithium-ion battery. We aim at computing the voltage as a function of the charge and examining the efficiency of the cell.

Model

As shown in the figure, we model the electrode as a lattice of occupied and unoccupied lithium sites between corner-sharing metal-oxide sites.

The state of the electrode is described by the array σij. The value σij = –1 means that the site is a polyhedron,

σij = 0 means that the site is empty, and σij = +1 means that the site is occupied by lithium.

Graphically, metal-oxide sites (σij = –1) are represented by grey cells, empty lithium sites (σij = 0) are represented by white cells, and filled lithium sites

(σij = +1) are represented by blue cells.

The energy model has three parameters, namely, the energy of each occupied site ε, the applied voltage ϕ, and the energy J of a lithium–vacancy neighboring pair. The potential energy of the system reads

where δx=y is a shorthand notation that means 1 when x = y and 0 otherwise, and e stands for the charge of an electron .

The energy unit is the electron-volt (eV), the voltage unit is the volt (V), and the charge unit is the electron’s charge, meaning that e can be replaced by 1 in the expression of U that is written above.

Using this model and Monte Carlo simulations we are going to calculate the electrical response of the battery during charge (starting from a filled electrode) and discharge (starting from on empty electrode).

Assignment

Question 1

Explain the Metropolis algorithm to generate a collection of states that fulfills a given probability distribution. (5 points)

Using Boltzmann’s law, show that the Metropolis selection rule from an old state of energy Uold to a new state of energy Unew can be written as

R < exp( – (Unew – Uold)/(kBT) )

where R is a random number between 0 and 1. (10 points)

Complete the code accordingly (line 44 in the .m). Show your work. (5 points)

Question 2

Complete the function potential_energy for the calculation of the energy of the electrode (lines 128 to 153 in the .m). (20 points)

Question 3

The function change_random (lines 155 to 168 in the .m) selects a lithium site at random and changes its state (from occupied to unoccupied or from unoccupied to occupied).

Copy the lines of this function in your report and explain what each line means. (20 points)

Question 4

Assuming that there is no interaction between the sites (J = 0 eV), draw the final states of the Monte Carlo evolution for the voltages ϕ = 3.3, 3.4, 3.5, 3.6, and 3.7 V at T = 500 K. (20 points)

Question 5

Draw the dependence of the voltage ϕ as a function of the final lithium fraction xLi for J = 0.02, 0.04, and 0.06 eV. In each case draw the charge curve (starting from filled electrode) and the discharge curve (starting from the empty electrode) from ϕ = 3.3 to 3.7 V with voltage steps of 0.01 V at T = 500 K. (15 points)

Hint. Your curves should look like the one below. (Do not worry about statistical errors.)

What is the effect of J on the efficiency of the battery, that is, the ratio of the supplied energy to the stored energy? A qualitative answer will be accepted. (5 points)

Extra-Credit Question

The curves shown above are not smooth due to statistical noise in the Monte Carlo method.

Propose a way to reduce statistical errors and demonstrate your method works by drawing more accurate charge–discharge curves. (15 points)