Metrics

No single LIB is capable of meeting all of the demands of the large variety of applica- tions for LIBs. Therefore individual applications and technologies can be targeted by changing the architecture and chemistries of LIBs. Depending upon which elements and architecture are chosen for a LIB, various performance metrics can be achieved at the cost of others. Therefore, it is necessary to have standard techniques to evaluate the performance of LIBs for practical application. For several of the performance criteria of electrochemical power sources, a distinc- tion must be made between the theoretical values and the practical values. Theoretical values are calculated from the thermodynamics of the electrochemical cell reaction. Practical values are related to the total mass of the complete battery, including the mass of the electrolyte, the separa- tors, the current collectors, the terminals, and the cell housing [4]. In this work, the specific capacity always refers to the weight of the active material component only. Key performance indicators are summarized next.

2.1.2.1 Standard free energy and electromotive force

Reactions take place at both the cathode and the anode in the reaction sites situated at electrode-electrolyte interfaces of Li-ion cells. Thermodynamically, the reduction reaction at one electrode can be represented by

aA + ne cC Eq. (2.9)

where a molecules of A take up n electrons e to form c molecules of C. Simultaneously the other electrode undergoes oxidation that can be described by

bB - ne dD Eq. (2.10)

The overall reaction in the cell is given by addition of these two half cell reactions.

aA + bB cC+dD Eq. (2.11)

ǻ*ƒ -nFE° Eq. (2.12)

where, F is the faraday constant, 96487 coulombs, and E° is the standard electromotive force. The free energy of the reaction from Eq. (2.11) is defined by

ǻ* ǻ*ƒ Eq. (2.13)

where, a, R, T is activity of relevant species, gas constant, and absolute temperature, respective- ly. Inserting Eq. (2.12) into Eq. (2.13) gives the cell voltage which is described by Eq. (2.14), known as the Nernst Equation.

E = E° - Eq. (2.14)

7KHFKDQJHLQWKHVWDQGDUGIUHHHQHUJ\ǻ*ƒRIDFHOOUHDFWLRQLVWKHGULYLQJIRUFHZKLFK enables a Li-ion battery to deliver electrical energy to an external circuit. Together with activity coefficients, equilibrium constants, and solubility products in the reaction processes, measure- ment of the electromotive force makes available data on changes in free energy, entropies and enthalpies. The standard electromotive forces of half reactions for either oxidation or reduction

are summarized in Table 2.2 with respect to the “zero” reference electrode H2/H+ [8]. As it is

impossible to measure individual electrode potentials in an absolute sense, they are each meas- ured with reference to another electrode, which is used as standard electrode. The electrode normally used for this purpose is the standard hydrogen electrode (SHE), where the temperature is equal to 25oC, the pressure is equal to 1 bar, and all species are at unity activity.

Table 2.2: Standard potentials in aqueous solutions at 25oC [5].

2.1.2.2 Theoretical potential

The standard potential of the LIBs is determined by the type of active materials contained in the battery and can be calculated from the electrode potentials (reduction potentials) of the

half reactions. The overall theoretical cell voltage, ǻEo _{is obtained by subtracting the negative}

electrode potential, Eo,(-)_{, from the positive electrode potential, E}o,(+)_:

(2.15)

The maximum DFFHVVLEOHHQHUJ\LVVLPSO\WKHIUHHHQHUJ\RIUHDFWLRQǻ*&RQVHTXHQtly, high energy results from the choice of electrode materials. This can be achieved by the selection of HOHFWURGHVZLWKWKHJUHDWHVWGLIIHUHQFHRIHOHFWURFKHPLFDOSRWHQWLDOȝ>@7Ke cell voltage can be derived from the standard Gibbs free energy (Eq. 2.12) of the equivalent chemical reaction reorganized as:

(2.16)

For example, in the reaction of a Li metal/graphite cell, the theoretical voltage is calcu- lated as the sum of the oxidation potential of the reaction from Li metal to Li cation and electron

(=3.08V) and the reduction potential of the reaction from carbon, Li cation, and electron to LiC6

(= -2.80V). According to Eq. (2.5) and (2.6), the theoretical voltage of Li metal/graphite cell is 0.28V.

Anode: Li Li++ e- Eoxidation = -(-3.08) V Eq. (2.17)

Cathode: Li++ e-+ C6 LiC6 Ereduction = -2.80 V Eq. (2.18)

Figure 2.5 compares the electrochemical potential ranges of popular lithium insertion compounds vs. lithium metal schematically, with cell voltages as high as 5 V vs. lithium metal possible with variations of Li1-xCoMnO4and Li1-xNi0.5Mn1.5O4.

Figure 2.5: Electrochemical potential rages of common lithium insertion compounds in reference to metallic lithium [10].

2.1.2.3 Theoretical capacity

The theoretical capacity of an electrochemical cell is based only on the amount of active material present and participating in the electrochemical reaction, calculated from the equivalent weight of the reactants. It is expressed as the total quantity of electricity involved in the electro- chemical reaction and is defined in terms of coulombs or ampere-hours (Ah). The ampere-hour capacity of a battery is directly associated with the quantity of electricity obtained from the active materials. Theoretically 1 gram-equivalent weight of material will deliver 96,487 C or 26.8 Ah. In general, 1 gram-equivalent weight is the atomic or molecular weight of the active material in grams divided by the number of electrons involved in the reaction. Similarly, the ampere-hour capacity on a volume basis can be calculated using the appropriate data for ampere-

hour per cubic centimeter. The theoretical specific charge capacity, qth (Ah kg-1), can be ex-

pressed by the amount of charge per kg of reactants, mi, calculated from the stoichiometry of the overall cell reaction (Eq. 2.19):

(2.19)

2.1.2.4 Energy density and power density

The capacity of a LIB can also be expressed in terms of energy, taking both the voltage and the quantity of electricity into consideration. This theoretical energy value is the maximum value that can be delivered by a specific LIB system and it is calculated as the product of voltage (V) and capacity (Ah) into units of watt-hour (Wh):

(Wh) = (V) x (Ah) (2.20)

Energy density is calculated as the energy per unit mass (Wh kg-1) and is expressed with respect

to either the weight of the active material only, the composite electrode weight, the weight of the inter-battery components (anode, cathode, electrolyte/separator), or the entire cell with packag- ing included. The value for energy density decreases with additional weight of inactive components of a battery and is therefore always less than the theoretical energy density. Addi- tionally, there are inevitable energy losses that become more significant at high charge discharge rate.

Power is defined as the rate at which energy is delivered from or to a battery, the maxi- mum power of which a battery delivers or accepts is determined by the highest rate at which the LIB can be cycled without any failure. The maximum power is governed by kinetic processes of the complete battery system. In practice, the charge/discharge rate is expressed in terms of C- rate and defined as:

I = M Cn Eq. (2.21)

where, I = dis(charge) current, [A]

n = time for which rated capacity is declared, [hr]

M = multiple or fraction of C

For example, C/37 refers to a dis(charge) time of 37 hours, or a full cycle time of 74 hours based on the amount and theoretical capacity of the active material. Likewise, 10C represents a cell being dis(charged) in 6 minutes, with a total discharge-charge time of 12 minutes. With increas- ing active material capacity, the absolute current being applied will increase while the C-rate stays constant, so care must be taken in consideration of comparing various materials. The power (W) can then be calculated as:

(W) = (V) x (Ah) x C-rate (hr-1) Eq. (2.22)

Similar to calculation of energy density, power density is calculated relative to the desired

components for the most meaningful result and can be expressed in watts per kilogram (W kg-1)

2.1.2.5 Cycling behavior and coulombic efficiency

In many applications a secondary battery is expected to maintain its major properties over many discharge/charge cycles. This can be a serious practical challenge, and is often given a lot of attention during development and optimization of rechargeable batteries. Cycling behavior depends on the coulombic efficiency, defined as the fraction of the prior charge capacity that is available during the following discharge. It can be observed that even minor inefficiency per cycle can have important consequences (Figure 2.6). For example, a half percent loss per cycle causes available capacity to drop to only 78% of the original value after 50 cycles. After 100 cycles only 61% remains at that rate. The cycling behavior degrades rapidly for even lower coulombic efficiency [5].

2.1.2.6 Self discharge

Another property of importance in practical cells is self-discharge, or, the reduction in available capacity with time, even without energy being taken from the LIBs by passage of current through the external circuit. This is also relevant to the shelf-life of practical cell or batteries and is a serious practical problem in some system of LIBs.

As previously discussed, the capacity is a property of the electrodes and its value at any time is determined by the remaining extent of the chemical reaction between the neutral species in the electrodes. Any self-discharge mechanism that reduces the remaining capacity must involve either transport of neutral species or concurrent transportation of neutral combinations of charged species, through the cell. The transport of neutral species can occur such that individual neutral species can move across from one electrode to the other such as transporting through an adjacent vapor phase, cracking the solid electrolyte, or dissolving the gas in a liquid electrode. Since the transport of charged species is not involved, these processes produce chemical self- discharge. Concurrent transportation of neutral combinations of charged species involves the transport of charged species, and therefore is called electrochemical self-discharging. Self discharge can also be a result of impurities within the constituent electrodes reacting with the electrolyte, reducing the available capacity over time.