Battery voltage and capacity in non-equilibrium

The battery voltage described by the Nernst Equation and battery capacity assumes that the battery is in equilibrium. Since a battery under load is not in equilibrium, the measured voltage and battery capacity may differ significantly from the equilibrium values, and the further from equilibrium (ie the high the charge or discharge currents), the larger the deviation between the battery voltage and capacity equilibrium and the realistic battery voltage may be. The difference between the voltage under equilibrium and that with a current flow is termed polarization. Polarization effects have a significant impact on battery operation, both beneficial and detrimental. For example, polarization effects mean that under normal operation of lead acid batteries the electrolysis of water proceeds slowly and to first order can be neglected during discharge (but not charging since the voltage is higher). However, polarization effects also have detrimental effects on performance, by, for example, reducing efficiency and making the battery capacity sensitive to charging and discharging conditions.

The polarization is comprised of three basic mechanisms, relating to resistive drops in the battery, and to two effects relating to the rates at which a reaction can proceed. These two effects are due to kinetic effects caused by the inherent rates of the chemical reaction (called kinetic overvoltage or activation overvoltage), and by the effects related to the movement of reactants to the electrode (called mass transport overvoltage). The overvoltage causes a deviation of the voltage and capacity from the equilibrium values calculated earlier. As shown below, during discharging, the battery voltage is lower than that in equilibrium, while during charging, a higher voltage than the Nernst voltage is required. Polarization effects have significant impact on the battery efficiency and how the battery can be charged and discharged.


Voltage vs. Current

Cell potential away from equilibrium and including polarization effects.

Reaction Rates and Polarization

The equilibrium electrochemical potentials only take into account the initial and final potentials of the materials in the reaction, without considering the rates or kinetics of the reactions themselves. The chemical reaction rates play an important role in determining the operation of a battery and in the processes that control battery behavior. For example, if multiple reactions can occur, then a reaction with a reaction rate significantly lower than all other reaction rates will not proceed to a significant extend and may potentially be ignored.

A single chemical reaction is typically composed of multiple steps, and each of these steps has a particular rate. The reaction rate is controlled by two processes. First, in order for the reaction to proceed, all the reactants must be physically present in one location, which for a battery is the electrode. The processes which involve the transport of the reactants in their appropriate form to the site of the chemical reaction are called mass transport or concentration overpotential. The steps in getting the reactants to the electrode are shown below: all the reactants must be present in their appropriate form (ie in solution or as a solid), those in solution must diffuse to the site of the reaction, the reactant species must absorb on the surface of the electrode (if the electrode is part of the chemical reaction), and finally the electron transfer must occur. Applied to a lead acid battery, this means that both the lead metal and the lead ion must be present. This involve the dissolution of the metallic ion (if it is present in solid form, as in the lead acid case shown below), the transport of the reactants from the electrolyte to the electrode surface, and the adsorption of the necessary components on the electrode surface.

In addition to the transport of the reactant species to the site of the reaction, a second possible rate limiting step for the reaction is the rate at which the chemical reaction proceeds due to the kinetics of the chemical reaction. In many chemical reactions, the reacting species form short-lived intermediate products, and then these intermediate products react to form the final products. If the rate of formation of the intermediate species is slower than the remaining steps, then these intermediate steps control the reaction rate. Further, the energy required to form these intermediate products may be higher than the average energy of the reactants. As the reactants have a distribution of kinetic energy, and only those with higher energy can form the intermediate products. In this case, only a fraction of the initial reactants have sufficient energy to allow the reaction to proceed, thus limiting the reaction rate. The higher energy of the intermediate species gives rise to an activation energy, as shown in the figure below. In order for the reaction to proceed at a rapid rate, the reactants must be given energy greater than the activation energy. As the kinetic energy of the reactants is determined by their temperature, increasing the temperature of the reactants is a simple (but for batteries often impractical or accompanied by numerous other negative aspects) way to increase the reaction rate and decrease the overpotential.

Another way to decrease the activation energy may be reduced for some reactions by the use of a catalyst. In some chemical reactions, the reactant atoms must interact or collide in a particular way, such that a new material forms. For example, the interaction may require that the reactants a physically oriented in a particular way, as shown in the figure below. For such reactions, the addition of other chemical species that tend to orient the molecules in a specific orientation increase the probability of the reaction proceeding. Materials that have such an effect are called catalysts. This effect makes reaction rates sensitive to the presence of a small number of other species, which do not appear in the formula of the chemical reaction.

Mass Transport Overvoltage in Batteries

The mass transport overvoltage has a significant impact on batteries, particularly at high rates of charge and discharge. As the battery discharges, it depletes the region around the electrode of some of the reactants. The concentration gradient between the region surrounding the electrode and further away in the electrolyte causes reactants to diffuse towards the electrode. However, if the discharge rate of the battery causes the reactants to be used at a greater rate than they can diffuse towards electrode, then the concentration near the electrode will continue to drop as the battery discharges. This drop in concentration is greater than that expected voltage drop if the reactants were uniformly distributed through the electrolyte and therefore, according to the Nernst equation, the battery voltage decreases more rapidly than that calculated by equilibrium. The more rapidly a battery is discharged, the more rapid the fall in voltage compared to that from equilibrium. Rapid discharging affects not only the battery voltage, but also battery capacity. Since some of the reactants are not used in the reaction before the voltage drops below the minimum voltage, then the available battery capacity is also reduced.

During charging, a similar process occurs, except that charging increases the concentration surrounding the electrode. Consequently, a higher voltage is required to charge the battery than expected by equilibrium calculations. The mass transport overvoltage has a significant effect on the battery parameters relevant to a photovoltaic system. The lower voltages during discharge and higher voltages during charging reduce the battery efficiency. Further, mass transport effects alter the available battery capacity, as the battery capacity is reduced under high discharge rates. Because of these effects, mass transport has a significant impact on the optimal use of a battery, limiting both the charge and discharge currents.

Activation Overvoltage in Batteries

The kinetic or activation overvoltage of the reduction and oxidation reactions of the battery should be as small as possible, since during charging the voltage required will greater than the equilibrium voltage by activation energy. The difference in the charging voltage and the discharging voltage (i.e., the overvoltage) reduces the battery efficiency.

If there are secondary or side reactions in the battery, then the kinetic overpotential has different effects between charging and discharging. During discharging, the battery voltage is lower, and therefore there is less possibility that the voltage is sufficient to overcome the activation energy of secondary battery reactions. During charging, the battery voltage is higher, and hence there is the possibility that additional reactions can occur. This effect can give rise to beneficial properties. The hydrolysis of water consists of the redox reaction shown below, which has an electrochemical potential of 1.23 V.

Consequently, if a voltage of more than 1.23V is applied to a battery which has water as a component of the electrolyte, then the electrolysis of water occurs, producing hydrogen and oxygen instead of the charging reaction for the battery. Since most batteries operate at about 2V, this would then make water-based electrolytes unsuitable for batteries. However, the overvoltage of the redox reactions in the electrolysis of water are high enough such that during discharging, gas evolution from the electrolysis of water (or either one of the half reaction involved in the electrolysis of water) is not a dominant consideration. However, during charging, the higher voltage experienced by the battery causes first the hydrogen and then the oxygen half reactions to proceed. In lead acid battery systems, the presence of these two reactions gives rise to gassing. In many battery configurations, gassing leads to numerous undesirable side-effects, including water loss from the electrolyte and physical damage to the electrolyte.

Resistive Drops in Batteries

A final contribution to the overvoltage in a battery is the resistive drops that occur in a battery. There are several components to the total battery resistance. Part of the overall resistance is due to resistance of the components in the path of the electron flow, including the electrode and the connections between the two electrodes. Other components of the resistive polarization include the surface of the electrodes. The resistive polarization may not be linear with applied voltage. Other components of the resistive polarization include the resistance of the surface of the electrode. For example, in a lead acid battery, as the discharge reaction proceeds, lead sulfate builds up on the surface of the electrode, which is non-conductive. The resistive overpolarization has several practical impacts on battery performance and operation. Similar to the concentration polarization, it reduces the efficiency and places limits on how much the battery can be charged or discharged.

Secondary Reactions

In addition to the central reduction and oxidation reaction which comprise a battery, secondary or side reactions may occur. In most cases, these side reactions give rise to unwanted or detrimental effects. In all cases, these secondary reactions reduce the coulombic efficiency of the battery. If the secondary reaction occurs during discharging, some of the charge (current that would normally flow to the load is used by the secondary reaction). Similarly, during charging, the secondary reactions use charge intended to drive the main battery reactions, thus reducing the couloumbic efficiency. The electrolysis of water described in the activation overpotential is an example of an unwanted secondary reaction. Secondary reactions give rise to several unwanted effects, such a gassing, self-discharge and corrosion of the electrodes.

Physical State of the Electrodes

The physical state of the electrodes plays an important part in the practical operation of a battery. The key characteristic of a battery electrode is that its surface area should be large. This lowers the series resistance, increases the area over which the chemical reaction can take place (hence also reducing the mass transport overvoltage). In addition, a large surface area helps ensure that the reactants are not completely covered by the products of the chemical reaction. A complete, uniform coverage of the electrode by the product reaction would prevent the redox reactions from proceeding, since the reactant species could no longer reach the electrode. Moreover, even in the reaction products allow the reactant species through, the reaction products are often not conductive, and therefore electrons evolved or required by the redox reactions could not pass through the reaction productions. A large surface area is typically achieved by using porous materials. The figure below shows the porous lead used in a lead acid battery.

Porous Lead

During charging and discharging, several processes can occur that change the structure or shape of the electrode. In most battery reactions, the electrode materials undergo a physical change during the discharge/charge cycle. The changes to the electrode, both physical changes as the original electrode material is re-formed and chemical changes of the materials on the electrodes give rise to numerous non-idealities. A key non-ideality is that the material may change its morphology, potentially during deposition of the reaction products on the electrode, but more commonly when the electrode material remains unchanged for long periods of time. For example, in lead-acid batteries, lead sulfate, which forms as the battery is discharged, may form large, relatively insoluble crystals over time. These large crystals are difficult to convert back into lead or lead oxide, and hence they reduce battery capacity if the battery is left in its discharged state.

Other effects that relate to the physical changes experienced by the electrode or electrolyte are that the reactant products seldom have the identical density as the reactants, and hence the electrode undergoes physical changes in its size. If the mechanical stresses are too large, the electrode material may flake off, hence permanently reducing capacity. The relative physical changes in size may be exacerbated at high or low temperatures, as density differences may increase as the temperate changes.

Finally, as the electrode material is re-formed during charging, the electrode may change its shape. In lead acid batteries, this is circumvented by the fact that the solubility of the lead ion Pb2+ is very low, and hence Pb2+ is rapidly converted to Pb in the close physical proximity to where it was dissolved, thus preventing significant changes of shape of the electrode. Alternately, either the products during discharging or the original battery material during charging may form so as isolate regions from charging or discharging, thus permanently reducing battery capacity.