*2.3. Electrolyte*

The ideal electrolyte of the battery is a transport medium to enable lithium ions to move without resistance between electrodes. The electrolyte forms a temporary chemical bond with the lithium ions as they move from one electrode to the other. When the lithium ion reaches the other electrode, the SEI acts as a filter, removing the electrolyte surrounding the lithium ion and permitting only the lithium ion to flow through the SEI and into the cathode or anode [23]. This phenomenon is particularly important at the anode because when the lithium ion is bonded with the electrolyte, it has a much bigger size than the expansion capability of the graphite electrode. If the lithium ion with the electrolyte intercalate into the anode, then it results in uneven expansion and cracking of the anode. This cracking of the anode exposes it to the electrolyte and results in a growth of SEI and irreversible compounds. Figure 1 shows the temporary bond that the lithium ion makes with the electrolyte as it moves through the electrolyte and the SEI filtering the electrolyte to enable proper intercalation into the graphite anode. However, when subjected to extreme operating conditions such as high C-rates and temperatures, the electrolyte begins to interact with the lithium ions being transported and forms permanent chemical compounds. Since batteries generate and store energy through electrochemical reactions, the temperature of operation has an important role in the kind of reaction that takes place. When subjected to high temperatures, the ethylene carbonate in the electrolyte solution reacts with lithium ions to form more SEI and ethylene gas as shown in Equation (1) [24]. The ethylene gas causes the expansion of the battery cell because there is no exhaust or outlet for the gas to escape. This expansion applies pressure on the electrode and causes it to crack and results in more electrode material being exposed to the electrolyte. With more electrolyte-electrode exposure, the degradation rate is increased and more electrolyte and electrode material are lost. Equation (1) is an example of one of the side reactions that takes place in the battery that results in SEI formation.

$$\underbrace{\left(2\,\text{(CH}\_{2}\text{)}\_{2}\text{CO}\_{3}\right)}\_{\text{ethylene carbonate}} + 2\,\text{e}^{-} + 2\,\text{Li}^{+} \rightarrow \underbrace{\left(\left(\text{CH}\_{2}\text{OCO}\_{2}\text{Li}\right)\_{2}\right)}\_{\text{SI}} + \underbrace{\left(\text{CH}\_{2}=\text{CH}\_{2}\uparrow\right)}\_{\text{(ethylene gas)}}.\tag{1}$$

Another degradation phenomena that affects the electrolyte is dissociation when subjected to either very high temperatures or potentials. When the electrolyte is subjected to very high temperatures or potential differences across it, the chemical compounds begin to break down. This breakdown of the chemicals results in the inability of the electrolyte to act as a transportation medium between the electrodes.

This paper simulates the degradation of the electrolyte by varying the electrochemical properties of the electrolyte: the salt diffusion coefficient and the transference number using the first principle-based 4 dimensional degradation model (4DM) as shown in Figure 2 [24]. The terminal voltage is studied based on the degradation of these parameters and conclusions are drawn in terms of its sensitivity to the degradation of the electrolyte. The electrolyte salt diffusion coefficient and the transference number are also interdependent. Their interdependence can be observed in Equation (3) where the change in the effective diffusion coefficient is directly proportional to the transference number of the electrolyte. When either of the parameters change, there will be a more severe impact on the rate of change of the terminal voltage of the battery. While the 4DM framework is capable of simulating the interdependencies between the different parameters of the electrolyte, this article focuses on the sensitivity of each individual component of the battery on the performance and voltage response. As a result, the interdependencies are not considered in this article and will be presented in a separate article.

**Figure 2.** 4DM used to simulate electrolyte degradation.

#### **3. Electrolyte Salt Diffusion Coefficient Degradation**

The salt diffusion coefficient of the electrolyte defines the maximum rate of diffusion that is possible in the electrolyte [25]. The diffusion coefficient's relationship with the temperature and electromotive force (EMF) applied is given by the Stokes–Einstein equation [26].

$$D = \mu k\_B T.\tag{2}$$

Using the conservation of mass equation based on Fick's Law when applied to the Pseudo 2D (P2D) model, gives,

$$\frac{\partial \left( \varepsilon\_{\varepsilon} c\_{\varepsilon} \right)}{\partial t} = \frac{\partial}{\partial \mathbf{x}} \left( D\_{\varepsilon}^{eff} \frac{\partial c\_{\varepsilon}}{\partial \mathbf{x}} \right) + \left( \frac{1 - t\_{+}^{0}}{F} \right) j^{Li}. \tag{3}$$

Given *Deff e* = *Depe* , Equation (3) can be written as

$$\frac{\partial(\varepsilon\_t c\_\varepsilon)}{\partial t} = \left(D\_t \varepsilon\_t^p \frac{\partial^2 c\_\varepsilon}{\partial x^2}\right) + \left(\frac{1 - t\_+^0}{F}\right) j^{Li}.\tag{4}$$

For a constant *jLi*, if *pe* decreases then *∂*(*ece*) *∂t* will decrease. Integrating and solving Equation (4) with respect to time gives:

$$
\epsilon\_t \epsilon\_\varepsilon = D\_\varepsilon \epsilon\_\varepsilon^p \frac{\partial^2 \epsilon\_\varepsilon}{\partial x^2} t + \int \left( \frac{1 - t\_+^0}{F} \right) j^{Li} \partial t. \tag{5}
$$

Integrating on both sides of Equation (5) with respect to *x* and solving, we get,

$$c\_{\varepsilon} = \frac{1}{\left(\varepsilon\_{\varepsilon}\mathbf{x}^{2} - D\_{\varepsilon}\varepsilon\_{\varepsilon}^{p}t\right)} \iiint \left(\frac{1 - t\_{+}^{0}}{F}\right) j^{Li} \partial t \partial \mathbf{x} \,\mathrm{d}\mathbf{x}.\tag{6}$$

Therefore, when *pe* decreases then (*ex*<sup>2</sup> − *Depe t*) increases and *ce* decreases. From (3) and (7), for constant *jLi*, the decrease in *ce* will cause an increase in *φe* to balance the equation.

$$
\frac{
\partial
}{
\partial \mathbf{x}}
\left(
\kappa^{eff} \frac{
\partial \phi\_{\varepsilon}
}{
\partial \mathbf{x}}
+
\kappa\_{D}^{eff} \frac{
\partial \ln c\_{\varepsilon}
}{
\partial \mathbf{x}}
\right) + j^{Li} = 0,\tag{7}
$$

$$
\kappa^{eff} \frac{\partial^2 \Phi\_\ell}{\partial \mathbf{x}^2} = -\kappa\_D^{eff} \frac{\partial^2 \text{lnc}\_\ell}{\partial \mathbf{x}^2} - j^{Li}. \tag{8}
$$

Integrating (8) gives:

$$\phi\_{\varepsilon} = -\frac{\mathbf{x}\_{D}^{eff}}{\mathbf{x}^{eff}} \text{lnc}\_{\varepsilon} - \frac{j^{\text{Li}} \mathbf{x}^{2}}{\mathbf{x}^{eff}} ,\tag{9}$$

$$
\Delta\phi\_{\mathbf{f}} = \frac{\kappa\_D^{eff}}{\kappa^{eff}} \ln \frac{c\_{\mathbf{c}\_2}}{c\_{\mathbf{c}\_1}}.\tag{10}
$$

The overpotential of the battery can be written as,

$$
\eta = \phi\_s - \phi\_c - \mathcal{U}.\tag{11}
$$

Since *η* is constant and the equilibrium potential, *U*, at any defined concentration is also constant, the only parameter that can vary to compensate for the change in *φe* is *φ<sup>s</sup>*. Thus,

$$
\Delta \eta = \Delta \phi\_{\mathbb{S}} - \Delta \phi\_{\mathbb{E}} - \Delta l I\_{\prime} \tag{12}
$$

$$0 = \Delta\phi\_s - \Delta\phi\_{c\prime} \tag{13}$$

$$
\Delta \phi\_{\mathfrak{s}} = \Delta \phi\_{\mathfrak{c}}.\tag{14}
$$

Therefore, when *φe* decreases, then *φs* will increase to keep *η* constant. With an increase in *φ<sup>s</sup>*, *Vt* will either increase or decrease based on the battery mode of operation (charging/discharging).

$$V\_t = \phi\_{s\_+} - \phi\_{s\_-} - \frac{R\_f}{A}I\_\prime \tag{15}$$

$$
\Delta V\_t = \Delta \phi\_{s\_+} - \Delta \phi\_{s\_-}.\tag{16}
$$

From Equations (14) and (16),

$$
\Delta V\_t = f(\Delta \phi\_t). \tag{17}
$$

Thus, from Equations (10) and (17) we get,

$$
\Delta V\_l = f \left( \frac{\kappa\_D^{eff}}{\kappa^{eff}} \ln \frac{c\_{\varepsilon\_2}}{c\_{\varepsilon\_1}} \right). \tag{18}
$$

From Equation (18), if *ce*1 is the initial lithium ion concentration in the electrolyte when the battery is designed then the change in terminal voltage to the decrease in lithium ion concentration in the electrolyte follows an exponential decrease.

Figures 3–5 show the terminal voltage response to the change in the electrolyte salt diffusion coefficient degradation. It can be observed that there is a decrease in the terminal voltage when the battery's electrolyte salt diffusion coefficient decreases. This is in correlation with the mathematical derivations obtained in Equations (6) and (18). Figure 6 shows that the response follows an exponential curve. This is obtained using the curve fitting toolbox in the Matlab. The R<sup>2</sup> fit is determined to be 0.9952.

$$
\Delta V\_t = -0.02406e^{(-4.001D\_t)}.\tag{19}
$$

**Figure 3.** Terminal voltage vs. time for change in electrolyte salt diffusion coefficient from 1.0 to 0.8.

**Figure 4.** Terminal voltage vs. time for change in electrolyte salt diffusion coefficient from 0.7 to 0.5.

**Figure 5.** Terminal voltage vs. time for change in electrolyte salt diffusion coefficient from 0.4 to 0.1.

43

**Figure 6.** Change of terminal voltage to transference number vs. electrolyte salt diffusion coefficient.

#### **4. Electrolyte Transference Number Degradation**

The electrolyte transference number defines the ratio of current carried by ions in an electrolyte to the total current flowing through the electrolyte [27]. Since in any electrolyte there are positive and negative ions, the transference number is usually between 0 and 1 [26]. In lithium ion batteries, the electrolyte ion transference number is around 0.3–0.4.

Using Equation (7) and the conservation of charge per Fick's Law as shown in Equation (21) we have,

$$\frac{\partial}{\partial \mathbf{x}} \left( \kappa^{eff} \frac{\partial \phi\_{\mathbf{c}}}{\partial \mathbf{x}} + \kappa\_{D}^{eff} \frac{\partial \ln c\_{\mathbf{c}}}{\partial \mathbf{x}} \right) + j^{Li} = 0,\tag{20}$$

$$\kappa\_D^{eff} = \frac{2RT\kappa^{eff}}{F} (t\_+^0 - 1) \left( 1 + \frac{\partial Inf\_{\pm}}{\partial lnc\_{\varepsilon}} \right). \tag{21}$$

For a fixed current being drawn or sent into the battery, *jLi* is a constant. Assuming that *∂lnf*± *∂lnce* = 0 we get,

$$
\kappa\_D^{eff} = \frac{2RT\kappa^{eff}}{F}(t\_+^0 - 1). \tag{22}
$$

Differentiating *κeff D* with respect to *t*0+ gives:

$$\frac{\partial \kappa\_D^{eff}}{\partial t\_+^0} = \frac{2RT\kappa^{eff}}{F}.\tag{23}$$

Therefore, with an increase in *t*0+, there is an increase in *κeff D* . If *κeff D* increases then using Equation (20), the only way to keep the equation equal to 0 for a constant *jLi* is that *∂∂x κeff ∂φe ∂x* must decrease. But *κeff* is a constant for any material. Therefore, *<sup>∂</sup>*<sup>2</sup>*φe ∂x*<sup>2</sup> must decrease. Thus, Equation (20) can be written as, 

$$\frac{\partial^2 \phi\_{\varepsilon}}{\partial \mathbf{x}^2} = \frac{\left(-\frac{2\mathcal{R}\mathbf{T}\kappa^{eff}}{F}(t\_+^0 - 1)\frac{\partial^2 \ln \varepsilon\_{\varepsilon}}{\partial \mathbf{x}^2}\right) - j^{Li}}{\kappa^{eff}}.\tag{24}$$

Integrating (24) gives,

$$\phi\_{\mathbf{c}}(\mathbf{x}) = -\frac{2RT}{F}(t\_+^0 - 1)\ln c\_{\mathbf{c}}(\mathbf{x}) - \frac{j^{Li}\mathbf{x}^2}{\kappa^{eff}}.\tag{25}$$

Therefore, from Equations (17) and (25), a conclusion that the variation in the terminal voltage is directly proportional to the change in the electrolyte potential can be drawn.

$$
\Delta \phi\_{\varepsilon}(\mathbf{x}) = -\frac{2RT}{F} (\Delta t\_{+}^{0}) \ln c\_{\varepsilon}(\mathbf{x}).\tag{26}
$$

From Equations (17) and (26), there is a direct relationship between the change in the transference number and the change in the terminal voltage. However, the relationship also has a negative slope.

$$
\Delta V\_t = f\left(-\frac{2RT}{F}(\Delta t\_+^0)\ln c\_c(\mathbf{x})\right). \tag{27}
$$

From Equation (6), we know that *ce* is a function of *t*0+. Therefore, Equation (27) can be rewritten as,

$$
\Delta V\_l = f(\Delta t\_+^0). \tag{28}
$$

From the curve fitting toolbox in Matlab, the smallest order of the polynomial that fits the function is a second order polynomial with an R<sup>2</sup> fit of 0.9982.

$$
\Delta V\_t = -0.003808 \Delta t\_+^0 + 0.02002 \Delta t\_+^0 - 0.02652. \tag{29}
$$

The transference number varies from 0.04 to 1.12. Figures 7–12 show the transference number varies from 0.1 to 2.8 because the base transference number is set to 0.4 and is being scaled between 0.04 and 1.12. With a decrease in the transference number, there is an increased impedance to lithium ion flow across the electrolyte. This increase in the resistance causes an increased potential drop across the electrolyte and results in the battery reaching its cut-off voltage sooner. Figure 13 shows the relationship between the change in terminal voltage to the change in transference number vs. the change in the transference number. This figure is consistent with the results obtained from (28).

**Figure 7.** Terminal voltage vs. time for change in transference number from 2.8 to 2.4.

**Figure 8.** Terminal voltage vs. time for change in transference number from 2.3 to 1.9.

**Figure 9.** Terminal voltage vs. time for change in transference number from 1.8 to 1.4.

**Figure 10.** Terminal voltage vs. time for change in transference number from 1.3 to 0.9.

**Figure 11.** Terminal voltage vs. time for change in transference number from 0.8 to 0.4.

**Figure 12.** Terminal voltage vs. time for change in transference number from 0.3 to 0.1.

**Figure 13.** Change of terminal voltage to transference number vs. transference number.

#### **5. Conclusions and Future Work**

There are three major types of side reactions, and they occur at the solid–electrolyte interface, the current collector–electrode interface and in the electrolyte. This article highlights the impact of electrolyte degradation on the performance of the battery and the physical manifestation that this degradation phenomenon has on the terminal voltage of the battery. There are two ways to represent the degradation of the electrolyte—loss of electrolyte salt concentration and change in transference number. The loss of electrolyte concentration increases the resistance to the flow of lithium ions across the electrolyte. The electrolyte loses lithium ions when the battery is subjected to harsh operating conditions such as high temperatures or charging/discharging rates. The lithium ions react with the organic solvents in the electrolyte to form irreversible chemical compounds as shown in Equation (1). The decrease in the transference number of the electrolyte symbolizes the loss of charge-carrying lithium ions or the increase in the electrons that are carrying charge across the electrolyte (i.e., self-discharge). From Equations (18) and (27), it can be seen that the terminal voltage has an exponential decrease for decrease in the concentration of lithium ions in the electrolyte while it follows a quadratic function with the transference number. This implies that the change in the terminal voltage is more sensitive to the decrease in lithium ion concentration in the electrolyte than to the variation in transference number. Depending on the rate of change of the voltage, it is possible to determine the kind of side reaction that is dominant in the electrolyte.

Future work involves performing more sensitivity analysis on the degradation of different physical battery components and determining the sensitivity of the change in terminal voltage and capacity to the change in component degradation.

**Author Contributions:** Conceptualization: M.-Y.C. and B.B.; Methodology: B.B.; Software: B.B.; Validation: B.B.; Formal Analysis: Bharat Balagpopal; Investigation: B.B.; Data Curation: B.B.; Writing Original Draft preparation: B.B.; Writing-review and editing: M.-Y.C.; Visualization: B.B.; Supervision: M.-Y.C.; Project Administration: M.-Y.C.; Funding Acquisition: M.-Y.C. and B.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially supported by National Science Foundation Award Number IIP—1500208.

**Conflicts of Interest:** The authors declare no conflict of interest.
