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Things covered:
1.
2.
3.
4.
5.
6.
Understanding the shape of the discharge curve
Open Circuit behavior
Constant voltage operation
Heat generation in cells
Smart charging of batteries
Side reactions in batteries
Understanding the shape of the discharge curve:
Let us go back to the example of the Daniell cell. For the case where the Zn2+ concentration is not
changing and only the Cu2+ is changing, we used Faraday’s law along with the Nernst equation to
describe how the voltage changes with time. In the previous lecture we converted this equation to one
of capacity and plotted the voltage change with the capacity of the electrode on discharge. This is
shown in the graph below:
The main message in this plot was that when the potential of a battery is plotted with respect to the
capacity, any change in the curves with current density is a result of deviations from equilibrium.
To understand why the potential drops off at the end of discharge, one needs to only examine the change
in the concentration of Cu2+ in the cell as the battery is discharged. This is plotted below at 5 different
capacities (denoted by the 5 blue circles in the above figure).
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In the figure the abscissa refers to the dimension of the cell with 1.0 denoting the electrode surface and 0
denoting the interface between the cathode and the anode chamber. At the start of discharge (0 Ah) the
whole cell is filled with a 1 M solution of Cu2+ . As the Cu2+ is slowly deposited to Cu via the reaction
Cu 2+ + 2e − ↔ Cu
the concentration of Cu2+ in solution decreases. The potential of this system is governed by the Nernst
equation, written as
Where
is the surface concentration (same as the bulk concentration in this figure). As the Cu2+
concentration decreases, the potential of the cell decreases. As discharge proceeds, all the Cu2+ is
consumed from the cell and the potential of the cell starts to drop, with a precipitous drop when the
concentration approaches zero. Note that, as derived, no account is taken for the fact that a
concentration gradient could build up in the cell due to the decrease in concentration at the electrode
surface.
Impact of concentration gradients:
One can account for the concentration variation in the cell by solving for Fick’s second law of diffusion
along with a boundary condition that relates the flux of the Cu2+ concentration to the current. Doing so
and combining this equation with the Nernst equation above, results in a plot that more closely
resembles discharge curves of batteries:
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At very low currents, the shape of the curves resembles that of the equilibrium curve shown above. As
the current density is increased, two effects are seen (i) the voltage goes lower as the current is increased
and (ii) the capacity of the cell decreases.
As the current density is increased, the concentration of Cu2+ at the electrode surface decreases. If the
time for transport is larger than the time for reaction, then the concentration gradient starts to build up
between the electrode surface and the bulk of the solution. This is clearly seen by examining the
concentration across the cell for the highest current density in the above figure for the various capacities
given by the blue circle.
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As in the previous case the concentration is uniform at the start of the experiment. As discharge
proceeds, the concentration at the surface drops faster than the bulk concentration as a consequence of
the slow time constant for transport. From the Nernst equation it is clear that as the surface
concentration decreases, the potential also decreases.
As discharge proceeds, the surface concentration ultimately decreases to zero, resulting in the
precipitous drop in the cell voltage. Note that at the end of discharge (35 Ah), while the surface
concentration is zero, a significant amount of Cu2+ still remains in the cell. However, this is not
accessible because the rate of consumption of Cu2+ via reaction is faster than that supplied via transport.
This leads to a lower utilization in the cell.
Stirring the solution would allow the Cu2+ to reach the electrode surface and allows for greater
utilization.
Relaxing the cell at the end of the discharge would result in the concentration equilibrating through the
cell. A subsequent discharge at the slowest current density would result in complete deposition of all the
Cu2+ in the cell.
Note that while the above example ignores kinetic overpotential and ohmic losses, these will also play a
part in dictating the behavior of the cell.
Open Circuit Behavior of Batteries:
Let us now examine the open circuit behavior of batteries in more detail. As mentioned earlier, an open
circuit allows equilibration of the cell. In the example of the Daniell cell above, the relaxation of
interest is the concentration across the chamber. As the concentration changes, the potential of the cell,
as governed by the Nernst equation, changes (increases). This process continues until the concentration
profile becomes uniform across the cell. The time constant or this process can be estimated using
τ MT =
L2
D
Where L is the distance of the cell and D the diffusion coefficient of the diffusing species.
For a lead-acid battery the species of interest is the sulfuric acid electrolyte that diffuses across the
separator. With a typical separator thickness of 0.1 cm and a diffusion coefficient of 1×10-5 cm2/s, τMT
can be estimated to be 1000 s.
For a Li-ion cell, the species of interest is the Li+ ion that diffuses across the particle of the active
material. With a particle radius of 1 μm and a diffusion coefficient of 1×10-11 cm2/s, τMT can be
estimated to also be 1000 s.
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In real batteries ohmic, kinetic, and transport losses occur and consequently an open-circuit would result
in the dissipation of all these losses. Typically, ohmic losses dissipate in a very short time (<10 ms)
while kinetic losses can occur in the order of seconds.
The various time constants that occur in a real lead-acid battery are plotted in the Figure below where
the dissipation of the ohmic, kinetic, and mass transfer losses in the system is captured.
In real batteries, open circuit experiments are used in order to gauge the various resistances in the cell,
and also to allow dissipation of these losses.
τ Rct =
12.7
RT C
F ioj
Kinetic
Control
12.65
OCP
2
Mass transfer τ MT = L
DH +
Control
12.6
Voltage (V)
12.55
12.5
V2
12.45
12.4
Ohmic
12.35
τ ohmic =
a C L2
κ
12.3
V1
12.25
12.2
0
5
10
15
20
25
30
Time (s)
Constant Potential Operation:
Let us go back to the Cu deposition reaction in the Daniell cell. One can use the discharge curve as a
function of rate to make a Modified Peukert Plot, shown below.
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The plot shows that in order to completely utilize the battery (i.e., deposit all the copper) one needs to
perform a 500 h constant current discharge. At times less than 100h only a fraction of the copper is
deposited. While these numbers are given for discharge, one can imagine a similar long time constant
on charge, thereby limiting the applications where such batteries can be used.
One method to decrease the time to discharge (or charge) the battery is to use potential control, as
opposed to current control. In other words, one can impose a constant potential on the battery in order
to drive the reaction. For example, in the Daniell cell imposing a constant potential of say 0.1 V on the
cell would result in an overpotential of 1.0 V on the battery. The large overpotential for reaction results
in a significant reaction rate at the electrode surface resulting in a decrease in the concentration of Cu2+
in solution. As the Cu2+ concentration decreases, the overpotential also decreases (due to the decrease in
the equilibrium potential, as dictated by the Nernst equation). These changes can be tracked by plotted
the change in current vs. time for this system, shown below:
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Current (A)
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
Time (h)
The plot shows that the large driving force for reaction at short times results in a large current. The
current this decays exponentially with time as reaction proceeds. Note that in the constant current case,
a current of 1 A resulted in 60% utilization with a discharge time of 50 h. The consequence of the large
current is that the discharge reaction can be forced to occur in a short time. This is seen by plotted the
capacity of the battery as a function of time. The figure shows that the battery can be discharged to
close to its full capacity in less than 100 h. Compare this to the constant current case, where a time
greater than 150h was required to achieve the same capacity. This faster discharge has great
implications on charging of batteries.
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Capacity (Ah)
50
40
30
20
10
0
0
50
100
150
200
Time (h)
A second advantage of a constant potential experiment is that by controlling the potential one can
control the reactions that happen in the battery. In systems where a second reaction becomes more
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favorable at certain potentials, constant voltage allows the reaction of importance to proceed at the
maximum rate without complications arising from a second reaction. A good example of this is the
LiNi1/3Mn1/3Co1/3O2 material that we discussed two weeks ago. Below are charge curves on this system
at various C rates. One can see that as the charge rate increases, the battery is charged to a smaller
percent of its full capacity (~165 mAh/g). Notice the shape of the charge curve, especially at the end of
charge, when compared to the discharge curves. The characteristics sharp change in the slope is not
seen on charge. This is because, on charge, the sharp change in slope occurs at potential much higher
potentials. However, these potentials are inaccessible because of side reactions that become more
favorable.
A constant potential experiment allows the charging of this battery at rates that are faster than C/6, while
maintaining the potential below 4.3 V and avoiding the side reaction.
An example of this is a constant voltage charge on a graphite/LiCoO2 cell where 90% of the charge is
completed in 30 mins. This battery is typically used in cell phones and laptops.
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Source: Bob Spotnitz,
Heat Generation in Batteries:
When the potential of the battery deviates from the equilibrium value, heat generation occurs. In simple
terms the amount of heat generated can be estimated by
q = I (U − V ) in W/cm2 or J/s-cm2
where I in A/cm2 is negative on charge and positive on discharge. Using the heat generation one can
calculate the temperature increase in the battery by standard heat transfer equations.
The downside of performing a constant voltage charge is that the large currents generated result in large
amount of heat generation in the cell. For a typical 2 Ah Li-ion cell, the current at short times can be as
high as 10-15 A’s. This large current results in significant increase in heat as shown below
Source: Bob Spotnitz,
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In this example, the graphite/LiCoO2 cell phone battery is charge to 4.2 V from a fully discharged state
(at 380 min). The sudden increase in current results in a significant increase in temperature to ~ 90 C.
Such a large increase would be unacceptable in real cells.
Smart Charging of Batteries:
Smart charging algorithms allow the battery to be charged in a minimum amount of time, while ensuring
no side reactions or excessive heat generation. A commonly used method is termed CC/CV where CC
stands for constant current and CV stands for constant voltage. In this scheme, the cell is first charged at
a fast rate to the cutoff potential. Once the potential reaches the cutoff, the charging is changed to a
constant voltage charge. This scheme circumvents the problem with the CC charge (slow rates needed
to get 100% charging) and CV charging (excessive heat generation). A typical example in a cell phone
Li-ion battery is shown below where the battery is first charged at a C/2 rate for 1.5 h after which the
battery is changed to a CV step at 4.2 V for the next hour, resulting in a total charge time of 2.5 h.
Source: Bob Spotnitz,
The figure below shows the evolution of the charge during the two steps. During the CC step the battery
is charged to 80% of its useable capacity with the rest 20% coming from the CV step. A second case is
shown where the CC step is conducted at a higher current of 1C. In this case, the CC step is conducted
for 45 mins reaching 80% SOC with the rest coming from the CV step. The total charge time can be
reduced from 2.5 h to ~2 hours using the higher current. Note that the higher current results in a slightly
higher temperature in the battery.
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Source: Bob Spotnitz,
Battery charging algorithms vary from system to system and from manufacturer to manufacturer. In
addition to CC/CV charging, pulse charging (CC with an OCV step in between) and charging with either
a SOC limit, temperature limit, or pressure limit are common.
Side Reactions in Batteries:
As mentioned earlier, one criterion that limits the charging of Li-ion cells is the need to keep the
potential below 4.3 V on the positive electrode. At potentials above 4.3 V other reactions become
thermodynamically favorable. These reactions are detrimental to the life and safety of Li-ion cells. In
general, side reactions dictate many aspects of battery behavior.
In aqueous systems, water electrolysis provides the bounds for the stability of the electrolyte used. The
hydrogen evolution reaction via
2 H + + 2e − ↔ H 2
U0=0 V
and the oxygen evolution reaction via
H 2 O ↔ O2 + 4 H + + 4e −
U0=1.229 V
are a source of side reactions in lead-acid batteries.
In basic solutions, analogous reactions can be written, namely,
2 H 2 O + 2e − ↔ H 2 + 2OH −
4OH − ↔ O2 + 2 H 2 O + 4e −
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U0=-0.8277 V
U0=0.401
Irrespective of the pH of the solution, the potential window before water electrolysis occurs is always
1.22 V. Any electrode that has a potential above or below these potentials would be subject to this side
reaction. An electrode where the impact of the side reaction is clearly seen is the nickel hydroxide
electrode, used as a positive electrode in a Ni-MH cell. In this electrode, during charge, the nickel
hydroxide is converted to nickel oxyhydroxide (i.e. a proton is lost from the structure). The figure
below shows the charging of this battery with time where the equilibrium potential of this system is at
~350 mV with respect to a Ag/AgCl reference electrode. However, the equilibrium potential of the
oxygen reaction is at 0.22 V with respect to this reference in this solution. In other words, the potential
at which this battery operates is always above the thermodynamic stability of the electrolyte and so the
oxygen evolution reaction always occurs along with the main reaction. The oxygen reaction manifests
itself very clearly as the charge proceeds where the potential rises up slowly and reaches a second
plateau at ~450 mV. In this second plateau the reaction occurring is primarily the oxygen evolution
reaction.
Therefore only a part of the current will be used to perform useful work in charging the battery. The rest
is wasted in this side reaction. This results in lowered columbic efficiency between charge and
discharge defined as
η=
q disch arg e
q ch arg e
× 100
An efficiency of 100% means that no side reactions occur, while η<1 means a side reaction (in this case
an oxidative side reaction). Note that η>1 means that the side reaction is a reduction reaction.
Typically, positive electrode are subjected to oxidative side reactions and negative electrodes are
subjected to reductive side reactions.
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The plot below shows the discharge of the nickel electrode. While the electrode was charged for 2500 s,
the discharge of the battery lasted only 1700 s. The difference in capacity is the coulombs going toward
the side reaction. A classic sign of a side reaction is an efficiency which is less than 100%.
Charging the battery such that it does not reach the second plateau would help reduce the side reaction
and therefore increase η.
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