As is well known by any user of rechargeable batteries, performance degrades over time. Although a new battery might last for 3 or 4 hours, with extended use, the working time before recharging may now be just 1 or 2 hours. This reduction in operating time, also known as aging, represents a loss in the capacity of the cell and is typically quantified in ampere-hours. Loss of capacity is a combination of cycle (use) life and calendar life, since the battery can lose capacity even while being stored due to irreversible self-discharge as discussed in the previous section. Figure 7.12 shows the capacity fade of a typical lithium-ion cell. The capacity in A·h that is available before the cutoff voltage is reached decreases with increasing cycle number. The loss of capacity for the lithium-ion cell is in large part due to a decrease in the amount of cyclable lithium in the cell. Specifically, some of the lithium in the graphite negative electrode reacts with the electrolyte to form a solid electrolyte interphase (SEI). The lithium that reacts becomes unavailable for subsequent cycling.
Figure 7.12 Typical behavior for capacity fade of a lithium-ion cell.
SEI growth is just one of many causes of capacity fade in secondary lithium cells. The predominant mechanisms are different for other cell chemistries, but all secondary cells degrade with operation. Understanding the capacity fade and power fade is essential for designing battery systems. There are a number of mechanisms that can contribute to capacity fade. The first is loss of active material. SEI formation is such an example, where an irreversible side reaction consumes lithium from the cell. Active material can also be disconnected electrically from the electrode, typically due to volume changes associated with restructuring or even from excessive gassing in aqueous systems. In addition to capacity loss, larger polarization of the cell either from increased charge transfer or internal resistance will cause the cutoff potential to be reached sooner. This behavior effectively results in a lower useable capacity of the battery. Such increases in internal resistance with time are common for many battery systems. There can also be chemical shorts in cells. For instance, the transport of Ag from the positive to negative electrode in rechargeable silver cells, or the transport of lithium polysulfide in lithium–sulfur cells both represent such shorts (see Problem 7.28). Self-discharge, discussed in the previous section, can also contribute to reduction of battery life.
The end of life for a cell depends on the application, but batteries are often considered unusable when the capacity is reduced to less than 70 or 80% of rated capacity. A key factor that influences capacity fade is the depth of discharge or the extent to which a battery is discharged during cycling. Figure 7.13 shows the impact of DOD on the number of cycles that can be achieved, for example, lead–acid and lithium-ion cells. It is rare for most rechargeable batteries to be repeatedly cycled from a fully charged to fully discharged state. More often than not, the state of charge is maintained in a narrower window of say 40–70%. A principal reason for doing this is to reduce capacity fade.
Figure 7.13 Effect of depth of discharge on cycle life.
Temperature also has a strong effect on battery life. Sometimes higher temperatures initiate new mechanisms for failure, but for the most part the main impact is on accelerating existing failure modes.
Closure
In this chapter, a basic description of a battery has been presented and battery operation has been discussed. Cells are characterized as either primary or secondary, with the two main types of electrode reactions being reconstruction and insertion. Batteries are rated in terms of nominal voltage and ampere-hour capacity. Methods for calculating the theoretical capacity have been developed. The current–voltage response of the battery is described using the principles of thermodynamics, kinetics, and mass transfer. Other important phenomena that occur in batteries have been introduced: heat generation, self-discharge, and capacity fade. The information presented in this chapter will prepare the reader for the analysis and design of batteries found in Chapter 8.
Further Reading
Besenhard, J. O. and Daniel, C. (eds) (2011) Handbook of Battery Materials, Wiley-VCH Verlag GmbH.
Huggins, R.A. (2009) Advanced Batteries: Materials Science Aspects, New York, Springer.
Reddy, T. and Linden, D. (2010) Linden’s Handbook of Batteries, McGraw Hill.
Vincent, C.A. and Scrosati. B. (1997) Modern Batteries: An Introduction to Electrochemical Power Sources, Butterworth, Oxford.
Problems
7.1. Use data from Appendix A or Appendix C to determine values of Uθ for the:
A lead–acid battery (both lead and lead oxide react to form lead sulfate).
A zinc–air battery in alkaline media.
7.2. Sodium is far more abundant in the earth’s crust than lithium. Consequently, there is interest in replacing lithium as the negative electrode material with sodium in batteries. Consider the overall reaction of lithium with cobalt as an example for a new secondary battery:
Write the equivalent reaction where sodium replaces lithium. Categorize this reaction based on the discussion from Section 7.2.
Using the thermodynamic data provided in Appendix C, calculate the equilibrium potential, capacity in A·h·g−1, and specific energy for lithium and sodium versions of this battery.
7.3. A common primary battery for pacemakers is the lithium–iodine cell. The negative electrode is lithium metal, the positive electrode is a paste made with I2 and a small amount of polyvinylpyridine (PVP), and the separator is the ionic salt LiI. The overall reaction is
Write out the half-cell reactions and using the data from Appendix A, calculate the equilibrium potential and the theoretical capacity in A·h·g−1. You may treat the positive electrode paste as pure iodine. Categorize this reaction based on the discussion from Section 7.2.
7.4. The lithium–iodine cell described in Problem 7.3 is used for an implantable pacemaker. Note that LiI is produced during discharge, and this salt adds to the thickness of the separator. The nominal current is 28 μA and is assumed constant over the life of the cell. How much active material is needed for a 5-year life? At 37 °C, the LiI electrolyte has an ionic conductivity of 4 × 10-5 S·m-1. If the separator is formed in place from the overall reaction, and LiI has a density of 3494 kg·m-3, what is the voltage drop across the separator due to ohmic losses in the separator after 2.5 years? The cell area is 13 cm2. Please comment on the magnitude of the voltage drop. Is it important? Why or why not?
7.5. Most high-energy cells use lithium metal for the negative electrode; furthermore, rechargeable lithium systems rely on intercalation for reversible reactions at the cathode. Discuss the idea of replacing Li with Mg for future rechargeable cells. Specifically contrast and compare Li and Mg commenting on the following:
Specific capacity [A·h g-metal-1]
Volumetric capacity [A·h cm3-metal-1]
Earth abundance
Specific energy and energy density
Ionic radii
Charge/radius ratio (Hint: How is this likely to affect intercalation?)
7.6. The carbon monofluoride primary cell consists of a lithium metal negative electrode, and a carbon monofluoride CFx as the positive electrode. The carbon monofluoride is produced by the direct fluorination of coke or another carbon. The fluorine expands the carbon structure, creating a nonstoichiometric intercalation material; the value of x is about 1. The overall reaction is expressed as
If the equilibrium potential of this cell is about 3.0 V and x = 0.95, determine the theoretical specific energy of this battery. How does this value compare to the capacity of a commercial cell, which is about 450 W·h kg-1? Why are they different?
Calculate the theoretical specific energy of the lithium sulfur dioxide battery (Table 7.1) and compare it to that of the CFx cell.
7.7. Calculate the standard potential, Uθ, for the Ni/Fe (Edison cell) from the following information for the half-cell reactions.
7.8. Calculate the theoretical specific energy of the aluminum–air battery. The two electrode reactions are
7.9. Some implantable batteries must provide high-pulse power for short periods of time, a defibrillator for instance. This requirement that cannot be met with the Li/I2 cell (Problems 7.3 and 7.4). One battery for such a device is the lithium silver–vanadium–oxide cell (Li/SVO) cell. The overall reaction is
Ag2V4O11 is a highly ordered crystalline material consisting of vanadium oxide sheets alternating with silver ions. These layers persist with the lithiation of the material. The equilibrium potential is shown on the right. There are two plateaus followed by a sloping decrease in potential at x > 5. What does this behavior suggest about the phases of the products? Assuming that the potential must be greater than or equal to that of the second plateau, calculate the theoretical energy density and specific energy of this battery. Problem 7.12 explores power.
7.10. Li ions are shuttled between electrodes of a lithium-ion battery during operation. During the charging process, lithium ions are transported from the positive electrode to the negative electrode. For a binary electrolyte (lithium salt, LiX, in an organic solvent), sketch the concentration of the salt in the separator of the cell. Explain the profile and comment on how it would change with changes in the magnitude and/or direction of the current density, i.
7.11. For an ohmically limited battery, the potential of the cell is given by Vcell = U – IRint, where Rint is the internal resistance of the cell. Derive an expression for the maximum power. At what current and cell potential is the maximum power achieved? How are the results changed if there is a cutoff potential, Vco, below which operation of the cell is not recommended, that is reached first (i.e., before the maximum power)?
7.12. For the SVO cell (described in Problem 7.9), in addition to the low average power (∼50 μW), periodically a 1 W pulse is needed for 10 seconds. The resistance of the cell initially decreases but then increases as the reaction proceeds (see data in table). Assuming that the cell is ohmically limited and that the separator area is 0.0015 m2, at what value of x in LixAg2V4O11 is the cell no longer able to provide the required pulse power?
x U [V] Rint [Ω·cm2]
0.0 3.2 20
0.6 3.2 17
1.1 3.2 17
1.8 3.2 16
2.4 2.85 14
3 2.65 16
3.5 2.6 20
4.1 2.6 25
4.3 2.5 27
SVO data at 37 °C.
7.13. Fit the data for a small lead–acid cell with the Peukert equation and determine the value of k. Use the capacity at 1.09 C as the reference. How well does the model fit the data?
Rate [C] Capacity [Ah]
5.45 0.52
1.68 0.68
1.09 0.72
0.31 1.01
0.17 1.20
0.09 1.30
7.14. Fit the data (taken from Figure 7.6) to the Peukert equation and determine the value of k. Use the capacity at 1 C as the reference. How well does the model fit the data? What would be the effect of an increase in the temperature during discharge of the cell due to joule heating?
Rate [C] Capacity [Ah]
1 7.80
4.5 7.60
9 7.32
14 6.95
18 6.45
7.15. What is the theoretical specific capacity and energy for the Leclanché cell? The overall reaction is
Use 1.6 V as the average potential of the cell. A practical battery of this chemistry has a specific energy of about 85 W·h·kg−1. Please explain the difference between the theoretical and practical values.
7.16. Calculate theoretical specific energy for the lithium–air cell. Base your answer on the mass of lithium only. There are three possible reactions:
7.17. The effective conductivity of a 25 μm separator filled with electrolyte is measured to be 2 mS·m−1. The material has a porosity of 0.55. If the bulk conductivity of the electrolyte is 18 mS·m−1, what is the tortuosity of the separator material? When operating at current density of 4 A·m−2, calculate the ohmic loss in the separator.
7.18. A principal mechanism for power fade in lithium-ion cell is the irreversible reaction of lithium with the electrolyte to form the SEI. This rate of side reaction must be very small in order to have good cycle life. If the end of life is taken as when the capacity is 80% of initial capacity, what is the minimum coulombic efficiency to achieve 100 cycles? 1000 cycles?
7.19. During discharge of the lead–acid battery, the following reaction takes place at the positive electrode:
Discharge of a vertical electrode results in flow due to natural convection. Briefly explain the natural convection process on the electrode. Sketch the velocity profile near the positive electrode.
7.20. A lithium-ion battery is being discharged with a current density of i mA·cm−2. The positive electrode has a porous structure, and the electronic conductivity is much greater than the ionic conductivity, σ ≫ κ. Assume an open-circuit potential where U+ is essentially flat, but increases for high SOC and drops for low SOC.
Sketch the ionic current density, i2, across the separator and porous electrode at the start of the discharge.
Sketch the divergence of the current density; physically explain the shape of this curve.
Repeat (a) and (b) when the cell has nearly reached the end of its capacity. Again explain the shape.
How would the internal resistance change with depth of discharge for this cell?
7.21. Develop a simple model for growth of SEI formation in lithium-ion cells. Assume the rate-limiting step is the diffusion of solvent through the film. Show that the thickness of the film is proportional to the square root of time. Discuss how capacity and power fade would evolve under these conditions.
7.22. Starting with Equation 7.20, which gives the rate of heat generation in the absence of any side reactions or short circuits, develop an expression for the rate of heat generation as a function of current. Treat the cell as being ohmically limited with a resistance RΩ. You may also consider that the entropic contribution, , is constant. Finally, assume that there is an additional, constant rate of heat generation due to self-discharge, . Sketch the rate of heat generation as a function of current for the cell.
7.23. The rate of self-discharge is critical design parameter for primary batteries.
Assume that a primary battery is designed to last for 5 years at a constant average discharge rate. At what C-rate does this battery operate?
For this same cell, what is the equivalent C-rate for the self-discharge process if the current efficiency is to be kept above 90%? Assume that the self-discharge reaction operates as a chemical short in parallel with the main electrochemical reaction.
Because of the extremely long lives of some batteries, microcalorimetry is used to measure the rate of self-discharge. Data for a Li/I2 cell (described in Problem 7.3) are shown on the right. Estimate the current efficiency of the discharge. The equilibrium potential is 2.80 V, the cell resistance is 650 Ω, and the entropic contribution is 0.0092 J·C−1 at the cell temperature. Assume a nominal operating current of 70 μA.
I [μA] Q [μW]
0.047 5.95
14.5 5.79
31.3 6.63
56.5 7.97
125 15.02
7.24. During charging, oxygen can be evolved at the positive electrode of a lead–acid cell. In order to avoid adding water, this oxygen must be reduced back to water. In the so-called starved cell design, the electrolyte is limited so that there is some open porosity in the glass-mat separator. Therefore, oxygen can diffuse to the negative electrode. One set of proposed reactions at the negative electrode is
What is the net reaction? Describe how the evolution of oxygen at the positive electrode and its reaction at the negative electrode is in effect a shuttle mechanism with oxygen for the lead–acid cell. How does the oxygen reaction impact battery performance during charging? How does it impact performance during overcharge? In other words, what is the impact of overcharging these starved lead–acid cells? Finally, these cells are designed to be sealed from the atmosphere. What is the impact of having the cell open to the atmosphere on the rate of self-discharge of the starved cell?
7.25. It has been proposed that a small contamination of iron in the electrolyte can result in a shuttle mechanism of self-discharge of nickel–cadmium cells. What is the standard potential for this reaction?
The two electrode reactions for the NiCd cell can be represented by
Comment on the plausibility of such a self-discharge mechanism.
7.26. The discharge reaction for the lead–acid battery proceeds through a dissolution/precipitation reaction. The two reactions for the negative electrode are
This mechanism is depicted in the figure. A key feature is that lead dissolves from one portion of the electrode but precipitates at another. The solubility of Pb2+ is relatively low, around 2 g·m−3. How then can high currents be achieved in the lead–acid battery?
Assume that the dissolution and precipitation locations are separated by a distance of 1 mm. Using a diffusivity of 10−9 m2·s−1 for the lead ions, estimate the maximum current that can be achieved.
Rather than two planar electrodes, imagine a porous electrode that is also 1 mm thick and made from particles with a radius 10 μm packed together with a void volume of 0.5. What is the maximum superficial current here based on the pore diameter?
What do these results suggest about the distribution of precipitates in the electrodes?
7.27. An 8 A·h Ni–MH cell is charged and discharged adiabatically. The data for the temperature rise are shown in the figure (adapted from J. Therm. Anal. Calorim., 112, 997 (2013)). Explain the effect of rate on the temperature rise. Comparing the differences between the charging and discharging temperature rise, what can be inferred about the entropic contribution to heat generation? Notice that only during charging a bit above 30 °C, the temperature rise increases sharply. Your colleague suggests that the side reactions of oxygen evolution increase rapidly at high temperatures. Can this evolution and recombination explain the results?
7.28. The lithium–sulfur battery uses lithium metal for the negative electrode and sulfur with carbon for the positive electrode. The overall reaction is
The electrochemical process at the positive electrode goes through a series of sequential formation of lithium sulfides (Li2Sx), specifically
What are the half-cell reactions associated with this mechanism? The higher order polysulfides (x = 8, 6, 4) are soluble in the electrolyte. In contrast, Li2S2 and Li2S are much less soluble. How could this situation lead to self-discharge in these cells? Identify some options to mitigate this self-discharge.
7.29. The theoretical specific capacity of an electrode was introduced in Section 7.4. Of course, to make a full cell, a positive and negative electrode must be combined. If and represent the specific capacity of the positive and negative electrodes, show that the specific capacity of the full cell is given by
Here, it is assumed that the capacities in A·h of the two electrodes are the same; that is, the electrodes are matched. If the specific capacity of the positive electrode is 140 mA·h·g−1 and that of the negative electrode is 300 mA·h·g−1, what is the specific capacity of the full cell? If the specific capacity of the negative electrode were doubled to 1000 mA·h·g−1, how much improvement in the specific capacity of the full cell is achieved?
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