Chapter 4 - Kinetics and thermodynamics

Question 1

What are the possible kinetic limitations (of electron transport) to the current in a battery?

Answer 1

1) Electrons in metal contact (Ohmic resistance)
2) Electrons in a network of conductive additive (percolation and tunneling)
3) Electrons in active material (semiconductor/insulator/metal)
4) Electrons through binder / SEI (thermionic activation, tunneling, hopping transport)
5) Electrons over interfaces (Schottky barriers, defect assisted tunneling)
6) Electrons through separator (shouldn’t be any)
7) Electrons in electrolyte (shouldn’t be any)

Question 2

What are the possible kinetic limitations (of ion transport) to a current in a battery?

Answer 2

1) Ions in electrolyte (diffusion, drift, tortuosity)
2) Ions in active material (diffusion, nearest neighbour interactions, trapping, 2-phase reaction)
3) Ions over interfaces (thermionic activation, three-state reaction of electron/ion(atom)
4) Ions through SEI (hopping, multiple ionization levels)

Question 3

What is the equilibrium/open circuit voltage of a cell?

Answer 3

It is the difference in chemical potential between the electrodes.

Question 4

What is the overpotential of the cell?

Answer 4

The overpotential is the voltage change needed to go from equilibrium to actually seeing a current.

Question 5

What is the source of overpotential?

Answer 5

When we charge a battery, we add electrochemical energy inside the battery. Some of our effort is lost as heat.

Similarly, when the battery does work on the outer circuit, some of the electrochemical energy is lost as heat.

Question 6

What is hysteresis?

Answer 6

The hysteresis is the ‘history’ of the cell. When cycling a battery, we charge and discharge with constant current. The exact measured voltage is thus dependent on the history (are we charging or discharging?)

Question 7

What would hysteresis purists count as hysteresis?

Answer 7

Some would only count the systems where the reaction pathway is different in both directions. E.g. a 2-phase reaction in the active material.

Others limit the term to only describe systems where kinetic limitations in the material leads to different phases of the material being involved in the two reaction directions.

Question 8

What are the four different contributions to the overall overpotential?

Answer 8

1) Thermodynamic overpotential. Due to the activiation barrier. Will be slightly dependent on temperature, not on current.
2) Charge transfer overpotential. Exponential increase with current until it is no longer the limiting mechanism.
3) Ohmic overpoential. Gives a linear current-voltage curve until the current density is so high that…
4) Concentration overpoential. …. the ionic diffusion can no longer maintain the ion concentrations near the interface.

Question 9

What is the C-rate?

Answer 9

The C-rate is a materials specific current density. It is the current density where the entire capacity of the battery is spent in one hour.

Question 10

Describe the GITT / PITT technique.

Answer 10

GITT - Galvanostatic Intermittent Titration Technique.

We run the current at constant current for a short while (ie. a given charge), then relax the system and measure the equilibrium voltage.

PITT - Potentiostatic Intermittent Titration Technique

We increase the voltage stepwise. We then first get a rapid current increase, then it slows.

Question 11

How can we measure the loss associated with charge and discharge from GITT / PITT measurements?

Answer 11

The area under the curve vs. the equilibrium voltage profile would represent energy being lost for us.

Question 12

Explain dQ/dV

Answer 12

dQ/dV is the “density of states” from a galvanostatic plot. We plot the number of electrons extracted at the different voltage levels.

Question 13

Explain cyclic voltammetry

Answer 13

Gives the same visual representation of voltages where electrochemical activity occur, but is done with a constant voltage sweep rate (instead of constant current, as in dQ/dV).

Question 14

In a CV-plot there might be shifts of the peaks in the top and bottom lines. Why is this?

Answer 14

The different lines depend on the way we are cycling (lithiation / delithiation). The same peaks should be in both lines (for reversible reactions), but due to overpotential these peaks can be shifted relative to each other.

Question 15

Explain impedance spectroscopy

Answer 15

In impedance spectroscopy, we probe the response of the battery to AC-signals of different frequencies.

At different frequencies, different phenomena is measured. At very high frequencies, electron transport is measured (change is too fast for ions).

At very low frequency, electron transport is in a steady state, while ion transport limits the system.

We can model the battery as an equivalent circuit to fit the data to get information about resistance, capacitance and inductance in different sections of the battery. However, we can always overfit, so these results can be dubious.

Question 16

What can we say that temperature is a measure of, in terms of probabilities of occupying high energy states?

Answer 16

It is a measure of how unlikely it is that many low-energy states combine to give one high energy state.

Question 17

What is entropy?

Answer 17

Entropy is a way of counting the number of available system configurations. It is defined as the natural logarithm of the multiplicity.

Question 18

What is the multiplicity of the system?

Answer 18

The number of configurations of a given system.

Question 19

What is thermal equilibrium?

Answer 19

The situation when the flow of energy from electrons to phonons and from phonons to electrons is equally likely.

This is when the total number of states allowed by the energy distribution is at a maximum.

Question 20

Why is the entropy, defined as the logarithm of the multiplicity, a good definition for a quantity that determines the equilibrium?

Answer 20

Because it works out nicely mathematically. The equilibrium between two system occurs when the multiplicity is maximised, and the total derivative is 0:

g_tot = g_1*g_2

=> dg_1/dE * g_2 - dg_2/dE * g_1 = 0

=> 1/g_1 * dg_1/dE = 1/g_2 * dg_2/dE

Since the derivative of a logarithm, d ln(g)/dg = 1/g and then d ln(g(E))/dE = 1/g* dg/dE, we see we can rewrite it

d ln g_1 / dE = d ln g_2 / dE

Question 21

How is temperature defined from the entropy?

Answer 21

d ln g1 / dE = d S / dE = 1 / kT.

It can be interpreted as a typiucal unit of thermal energy. A phonon mode in a lattice or a kinetic or vibrational energy of a gas molecule.

Question 22

How much energy does kT correspond to at room temperature?

Answer 22

About 1/40 eV.

Question 23

How can we reach Boltzmann statistics from the definition of entropy and probabilities of finding an electron in a certain state?

Answer 23

If we have two states, a and b, the relative probability Pa/Pb of finding the electron in the two states is determined by how many different ways the phonons can rearrange in the two cases.

Pa/Pb = ga/gb = e^Sa / e^Sb = e^(Sa-Sb) = e^dS

If the energy difference between these are dE, then the entrop difference is:

dS = -dS/dE * dE = - dE/kT

So this gives us:

P(E + dE)/p(E) = e^-(dE/kT)