Stress generation and fracture in lithium insertion materials

Abstract

A mathematical model that calculates volume expansion and contraction and concentration and stress profiles during lithium insertion into and extraction from a spherical particle of electrode material has been developed. The maximum stress in the particle has been determined as a function of dimensionless current, which includes the charge rate, particle size, and diffusion coefficient. The effects of pressure-driven diffusion and nonideal interactions between the lithium and host material have also been described. The model predicts that carbonaceous particles will fracture in high-power applications such as hybrid-electric vehicle batteries.

Appendix

Thermodynamics and kinetics

For lithium insertion into a generic host material, we consider the following reversible electrochemical reaction:

$$ {\text{LiS}} \to {\text{Li}}^{ + } + {\text{e}}^{ - } + S, $$

(125)

where S represents the host material. The OCP, U, of the intercalation material relative to a lithium reference is given by the equation

$$ FU = \mu ^{0}_{{{\text{Li}}}} + \mu _{{\text{S}}} - \mu _{{{\text{LiS}}}} {\text{,}} $$

(126)

where F is Faraday’s constant and $\mu ^{0}_{{{\text{Li}}}}$ is the chemical potential of pure lithium metal, which does not depend upon the composition of the host material. As in Eq. (6), the chemical potential of a species can be expressed in terms of its mole fraction, activity coefficient, and secondary reference chemical potential:

$$ \mu _{i} = \mu ^{\theta }_{i} + RT\ln {\left( {\gamma _{i} x_{i} } \right)}. $$

(127)

Expanding the chemical potentials in Eq. (126), we have

$$ FU = FU^{\theta } + RT\ln {\left( {\frac{{x_{{\text{S}}} }} {{x_{{{\text{LiS}}}} }}} \right)} + RT\ln {\left( {\frac{{\gamma _{{\text{S}}} }} {{\gamma _{{{\text{LiS}}}} }}} \right)}, $$

(128)

where

$$ FU^{\theta } = \mu ^{0}_{{{\text{Li}}}} + \mu ^{\theta }_{{\text{S}}} - \mu ^{\theta }_{{{\text{LiS}}}} {\text{.}} $$

(129)

Differentiating Eq. (126) with respect to x _LiS yields

$$ F\frac{{\partial U}} {{\partial x_{{{\text{LiS}}}} }} = \frac{{\partial \mu _{{\text{S}}} }} {{\partial x_{{{\text{LiS}}}} }} - \frac{{\partial \mu _{{{\text{LiS}}}} }} {{\partial x_{{{\text{LiS}}}} }}. $$

(130)

The species S and LiS are not independent of each other; the amounts of the two species sum to a constant, and their chemical potentials are related by the Gibbs–Duhem equation:

$$ x_{{{\text{LiS}}}} d\mu _{{{\text{LiS}}}} + x_{{\text{S}}} d\mu _{{\text{S}}} = 0. $$

(131)

Thus, Eq. (130) becomes

$$ \frac{{\partial \mu _{{{\text{LiS}}}} }} {{\partial x_{{{\text{LiS}}}} }} = - F{\left( {1 - x_{{{\text{LiS}}}} } \right)}\frac{{\partial U}} {{\partial x_{{{\text{LiS}}}} }}. $$

(132)

Applying the analog of Eq. (7), the thermodynamic factor is therefore

$$ 1 + \frac{{\partial \ln \gamma _{{{\text{LiS}}}} }} {{\partial \ln x_{{{\text{LiS}}}} }} = \frac{{x_{{{\text{LiS}}}} }} {{RT}}\frac{{\partial \mu _{{{\text{LiS}}}} }} {{\partial x_{{{\text{LiS}}}} }} = - \frac{F} {{RT}}x_{{{\text{LiS}}}} {\left( {1 - x_{{{\text{LiS}}}} } \right)}\frac{{\partial U}} {{\partial x_{{{\text{LiS}}}} }}. $$

(133)

For an ideal solution,

$$ U = U^{\theta } + \frac{{RT}} {F}\ln {\left( {\frac{{1 - x_{{{\text{LiS}}}} }} {{x_{{{\text{LiS}}}} }}} \right)}, $$

(134)

and the thermodynamic factor is unity. The derivation shown here follows the development of Darling [32], and avoids invoking the excess free energy, which is used by Verbrugge and Koch to derive a similar relationship between the thermodynamic factor and OCP via interaction parameters [33]. Our approach is more general than that of Verbrugge and Koch, in that it accommodates the wide variety of empirical OCP fits available in the literature. However, care must be taken when the empirical fits do not correctly approach the dilute limits or when they result in negative values for the thermodynamic factor.

A kinetic model is required to determine the effect of nonidealities in the electrode and electrolyte upon the exchange current density. Typically, the concentrations of reactants and products are used, but here, we use their activities for the sake of generality. The current associated with reaction Eq. (125) (i.e., lithium insertion into a generic host material) can be expressed as

$$ i = F{\left[ {k_{{\text{f}}} \prime a_{{{\text{LiS}}}} \exp {\left( {\frac{{{\left( {1 - \beta } \right)}FV}} {{RT}}} \right)} - k_{{\text{b}}} \prime a_{{\text{S}}} a_{{{\text{Li}}^{ + } }} \exp {\left( {\frac{{ - \beta FV}} {{RT}}} \right)}} \right]}, $$

(135)

where a _LiS and a _S are activities of lithium-filled and lithium-void sites in the host material, respectively; a _Li+ is the activity of lithium ions in solution; $k_{{\text{f}}} \prime$ and $k_{{\text{b}}} \prime$ are the forward and backward rate constants, respectively; and V is the difference between the electrode potential and the potential in the solution near the electrode, relative to a lithium reference electrode in contact with electrolyte of a particular lithium ion concentration (e.g., 1 M).

The thermodynamic OCP is obtained by setting the current to zero:

$$ k_{{\text{f}}} \prime a_{{{\text{LiS}}}} \exp {\left( {\frac{{{\left( {1 - \beta } \right)}FU\prime }} {{RT}}} \right)} = k_{{\text{b}}} \prime a_{{\text{S}}} a_{{{\text{Li}}^{ + } }} \exp {\left( {\frac{{ - \beta FU\prime }} {{RT}}} \right)}{\text{.}} $$

(136)

We emphasize that U′ is different from the measured OCP given in Eq. (128), which is independent of the activity of lithium ions. In describing solid diffusion phenomena in the electrode, the activity of lithium ions is irrelevant and can be set arbitrarily to the reference activity.

We can rearrange Eq. (136) to solve for the forward rate preexponential factor:

$$ k_{{\text{f}}} \prime a_{{{\text{LiS}}}} = k_{{\text{b}}} \prime a_{{\text{S}}} a_{{{\text{Li}}^{ + } }} \exp {\left( {\frac{{ - FU\prime }} {{RT}}} \right)}{\text{.}} $$

(137)

Substituting this expression into Eq. (135), we have

$$ i = Fk_{{\text{b}}} \prime a_{{\text{S}}} a_{{{\text{Li}}^{ + } }} \exp {\left( {\frac{{ - \beta FU\prime }} {{RT}}} \right)}{\left[ {\exp {\left( {\frac{{{\left( {1 - \beta } \right)}F\eta _{{\text{s}}} }} {{RT}}} \right)} - \exp {\left( {\frac{{ - \beta F\eta _{{\text{s}}} }} {{RT}}} \right)}} \right]}{\text{,}} $$

(138)

where

$$ \eta _{{\text{s}}} = V - U\prime $$

(139)

is the surface overpotential of the reaction. Equations (137) and (138) can subsequently be combined to yield

$$ i = F{\left( {k_{{\text{f}}} \prime a_{{{\text{LiS}}}} } \right)}^{\beta } {\left( {k_{{\text{b}}} \prime a_{{\text{S}}} a_{{{\text{Li}}^{ + } }} } \right)}^{{1 - \beta }} {\left[ {\exp {\left( {\frac{{{\left( {1 - \beta } \right)}F\eta _{{\text{s}}} }} {{RT}}} \right)} - \exp {\left( {\frac{{ - \beta F\eta _{{\text{s}}} }} {{RT}}} \right)}} \right]}{\text{.}} $$

(140)

The activities of the solid species are given by

$$ a_{i} = a^{\theta }_{i} \gamma _{i} x_{i} {\text{,}} $$

(141)

where $a^{\theta }_{i}$ is the activity at a particular secondary reference state (i.e., it is a function of temperature and pressure, but not of composition), and γ _i is the mole fraction activity coefficient. For lithium ions, we use an activity that is independent of electrical state. That is,

$$ a_{{{\text{Li}}^{ + } }} = a^{\theta }_{{{\text{Li}}^{ + } ,n}} f_{{{\text{Li}}^{ + } ,n}} c_{{{\text{Li}}^{ + } }} {\text{,}} $$

(142)

where $a^{\theta }_{{{\text{Li}}^{ + } ,n}}$ and $f_{{{\text{Li}}^{ + } ,n}}$ are, respectively, the reference activity and activity coefficient of lithium ions with respect to a reference species, n, in solution (see Eqs. (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19) of reference [34]). The reference species must be ionic, and, in our binary electrolyte, is limited to lithium ions and the corresponding anion. Because the reference electrode used here is lithium metal, lithium ions are the natural choice for the reference ion. Accordingly,

$$ f_{{{\text{Li}}^{ + } ,n}} = \frac{{f_{{{\text{Li}}^{ + } }} }} {{f^{{{\text{ref}}}}_{{{\text{Li}}^{ + } }} }} $$

(143)

where $f^{{{\text{ref}}}}_{{Li^{ + } }}$ is the ionic activity coefficient at the reference concentration of lithium ions (i.e., the concentration adjacent to the reference electrode). The ratio does not depend upon an ill-defined electrical state, and therefore, neither does the activity of lithium ions used in the kinetic model. Note that with this choice of reference ion, $a^{\theta }_{{{\text{Li}}^{ + } ,n}} = 1$.

From the above definitions of activity, Eqs. (138) and (140) can be expressed as

$$ i = Fk_{{\text{b}}} \gamma _{{\text{S}}} x_{{\text{S}}} f_{{{\text{Li}}^{ + } ,n}} c_{{{\text{Li}}^{ + } }} \exp {\left( {\frac{{ - \beta FU\prime }} {{RT}}} \right)}{\left[ {\exp {\left( {\frac{{{\left( {1 - \beta } \right)}F\eta _{{\text{s}}} }} {{RT}}} \right)} - \exp {\left( {\frac{{ - \beta F\eta _{{\text{s}}} }} {{RT}}} \right)}} \right]} $$

(144)

and

$$ i = Fk^{\beta }_{{\text{f}}} k^{{1 - \beta }}_{{\text{b}}} \gamma ^{\beta }_{{{\text{LiS}}}} \gamma ^{{1 - \beta }}_{{\text{S}}} x^{\beta }_{{{\text{LiS}}}} x^{{1 - \beta }}_{{\text{S}}} {\left( {f_{{{\text{Li}}^{ + } ,n}} c_{{{\text{Li}}^{ + } }} } \right)}^{{1 - \beta }} \times {\left[ {\exp {\left( {\frac{{{\left( {1 - \beta } \right)}F\eta _{{\text{s}}} }} {{RT}}} \right)} - \exp {\left( {\frac{{ - \beta F\eta _{{\text{s}}} }} {{RT}}} \right)}} \right]}{\text{,}} $$

(145)

where $k_{{\text{f}}} = k_{{\text{f}}} \prime a^{\theta }_{{{\text{LiS}}}}$ and $k_{{\text{b}}} = k_{{\text{b}}} \prime a^{\theta }_{{\text{S}}}$. Equation (21) is the familiar Butler–Volmer equation, with the preexponential group constituting the exchange current density.

Solving for U′ in Eq. (145), using the Eqs. (141) and (142) for the activities, we have

$$ U\prime = U\prime ^{\theta } + \frac{{RT}} {F}\ln {\left( {\frac{{x_{{\text{S}}} }} {{x_{{{\text{LiS}}}} }}} \right)} + \frac{{RT}} {F}\ln {\left( {\frac{{\gamma _{{\text{S}}} }} {{\gamma _{{{\text{LiS}}}} }}} \right)} + \frac{{RT}} {F}\ln {\left( {f_{{{\text{Li}}^{ + } ,n}} c_{{{\text{Li}}^{ + } }} } \right)}{\text{,}} $$

(146)

where

$$ U\prime ^{\theta } = \frac{{RT}} {F}\ln {\left( {\frac{{k_{{\text{b}}} }} {{k_{{\text{f}}} }}} \right)}. $$

(147)

In contrast to the reference potential given in Eq. (128), this term is complicated by the fact that the ratio k _b/k _f has units of inverse molarity. Essentially, it is the reference potential from Eq. (128), with the dependence upon lithium ion concentration removed. This is why we distinguish between the thermodynamic U′ and the measured U; the latter has no ionic composition dependence. Section 5.7 of reference [34] elaborates further upon this subject.

Presumably, the OCP is Nernstian in the limit of small LiC₆ concentration. That is,

$$ {\mathop {\lim }\limits_{x_{{{\text{LiS}}}} \to 0} }U\prime = U\prime ^{\theta } + \frac{{RT}} {F}\ln {\left( {\frac{{x_{{\text{S}}} }} {{x_{{{\text{LiS}}}} }}} \right)} + \frac{{RT}} {F}\ln {\left( {f_{{{\text{Li}}^{ + } ,n}} c_{{{\text{Li}}^{ + } }} } \right)}. $$

(148)

Let g(x _LiS) represent the deviation from Nernstian behavior over the entire stoichiometric range of the material. In other words, let

$$ U\prime = U\prime ^{\theta } + \frac{{RT}} {F}\ln {\left( {\frac{{x_{{\text{S}}} }} {{x_{{{\text{LiS}}}} }}} \right)} + \frac{{RT}} {F}\ln {\left( {f_{{{\text{Li}}^{ + } ,n}} c_{{{\text{Li}}^{ + } }} } \right)} + g{\left( {x_{{{\text{LiS}}}} } \right)}. $$

(149)

Comparing Eq. (149) with Eq. (146), we see that

$$ g{\left( {x_{{{\text{LiS}}}} } \right)} = \frac{{RT}} {F}\ln \frac{{\gamma _{{\text{S}}} }} {{\gamma _{{{\text{LiS}}}} }}. $$

(150)

From the Gibbs–Duhem equation [Eq. (131)] and the relationship between the chemical potential of a species and its thermodynamic factor [Eq. (7)], we can show that the thermodynamic factor for species S is identical to the thermodynamic factor for species LiS. Hence, by Eq. (133), we have

$$ 1 + \frac{{\partial \ln \gamma _{{\text{S}}} }} {{\partial \ln x_{{\text{S}}} }} = - \frac{F} {{RT}}x_{{{\text{LiS}}}} {\left( {1 - x_{{{\text{LiS}}}} } \right)}\frac{{\partial U^{'} }} {{\partial x_{{{\text{LiS}}}} }}. $$

(151)

Here, we have made use of the fact that the dependence of U′ upon x _LiS is identical to that of U.

In defining our reference state, we specify that the activity coefficient of a species approaches unity in the limit of dilute lithium (i.e., vacant host material):

$$ \gamma _{i} \to 1\,{\text{as}}\,x_{{{\text{LiS}}}} \to 0. $$

(152)

Thus, rearranging and integrating (151) yields:

$$ \gamma _{{\text{S}}} = \frac{1} {{1 - x_{{{\text{LiS}}}} }}\exp {\left( { - \frac{{F{\left[ {{\int_{{\text{ }}0}^{{\text{ }}x_{{{\text{LiS}}}} } {U\prime {\left( x \right)}dx} } - x_{{{\text{LiS}}}} U\prime {\left( {x_{{{\text{LiS}}}} } \right)}} \right]}}} {{RT}}} \right)}{\text{.}} $$

(153)

Insertion of this result into Eq. (144) yields

$$ i = Fk_{{\text{b}}} f_{{{\text{Li}}^{ + } ,n}} c_{{{\text{Li}}^{ + } }} \exp {\left( { - \frac{{F{\left[ {{\int_{{\text{ }}0}^{{\text{ }}x_{{{\text{LiS}}}} } {U\prime {\left( x \right)}dx} } + {\left( {\beta - x_{{{\text{LiS}}}} } \right)}U\prime {\left( {x_{{{\text{LiS}}}} } \right)}} \right]}}} {{RT}}} \right)} \times {\left[ {\exp {\left( {\frac{{{\left( {1 - \beta } \right)}F\eta _{{\text{s}}} }} {{RT}}} \right)} - \exp {\left( {\frac{{ - \beta F\eta _{{\text{s}}} }} {{RT}}} \right)}} \right]}{\text{.}} $$

(154)

The resulting exchange current density is a function of electrode composition via the OCP:

$$ i_{0} = Fk_{{\text{b}}} f_{{{\text{Li}}^{ + } ,n}} c_{{{\text{Li}}^{ + } }} \times \exp {\left( { - \frac{{F{\left[ {{\int_{{\text{ }}0}^{{\text{ }}x_{{{\text{LiS}}}} } {U\prime {\left( x \right)}dx} } + {\left( {\beta - x_{{{\text{LiS}}}} } \right)}U\prime {\left( {x_{{{\text{LiS}}}} } \right)}} \right]}}} {{RT}}} \right)}{\text{.}} $$

(155)

Substituting Eq. (149) into (155), we can write the exchange current density in terms of the nonideal portion of the OCP:

$$ i_{0} = Fk^{{1 - \beta }}_{{\text{b}}} k^{\beta }_{{\text{f}}} f^{{1 - \beta }}_{{{\text{Li}}^{ + } ,n}} c^{{1 - \beta }}_{{{\text{Li}}^{ + } }} {\left( {1 - x_{{{\text{LiS}}}} } \right)}^{{1 - \beta }} x^{\beta }_{{{\text{LiS}}}} \times \exp {\left( { - \frac{{F{\left[ {{\int_{{\text{ }}0}^{{\text{ }}x_{{{\text{LiS}}}} } {g{\left( x \right)}dx} } + {\left( {\beta - x_{{{\text{LiS}}}} } \right)}g{\left( {x_{{{\text{LiS}}}} } \right)}} \right]}}} {{RT}}} \right)}{\text{.}} $$

(156)

In dimensionless form,

$$ I_{0} = {\left( {1 - x_{{{\text{LiS}}}} } \right)}^{{1 - \beta }} x^{\beta }_{{{\text{LiS}}}} \Gamma {\left( {x_{{{\text{LiS}}}} } \right)}{\text{.}} $$

(157)

where

$$ I_{0} = \frac{{i_{0} }} {{Fk^{{1 - \beta }}_{{\text{b}}} k^{\beta }_{{\text{f}}} f^{{1 - \beta }}_{{{\text{Li}}^{ + } ,n}} c^{{1 - \beta }}_{{{\text{Li}}^{ + } }} }}, $$

(158)

$$ \Gamma {\left( {x_{{{\text{LiS}}}} } \right)} = exp{\left[ { - {\int_{{\text{ }}0}^{{\text{ }}x_{{{\text{LiS}}}} } {\overline{g} {\left( x \right)}dx} } - {\left( {\beta - x_{{{\text{LiS}}}} } \right)}\overline{g} {\left( {x_{{{\text{LiS}}}} } \right)}} \right]}{\text{,}} $$

(159)

and

$$ \overline{g} = \frac{{Fg}} {{RT}}. $$

(160)

For potentiostatic lithium extraction and insertion, the galvanostatic boundary condition Eq. (105) is replaced with

$$ {\text{N}}_{{{\text{LiS}}}} - \theta x_{{{\text{LiS}}}} \frac{{d\chi }} {{d\tau }} = \delta {\left( {1 - x_{{{\text{LiS}}}} } \right)}^{{1 - \beta }} x^{\beta }_{{{\text{LiS}}}} \Gamma {\left( {x_{{{\text{LiS}}}} } \right)} \times {\left\{ {\exp {\left[ {{\left( {1 - \beta } \right)}{\rm H}} \right]} - \exp {\left( { - \beta {\rm H}} \right)}} \right\}}\,{\text{at}}\,\xi = 1{\text{,}} $$

(161)

where

$$ {\text{H}} = \frac{{F\eta _{{\text{s}}} }} {{RT}} $$

(162)

and

$$ \delta = \frac{{k^{{1 - \beta }}_{{\text{b}}} f^{{1 - \beta }}_{{{\text{Li}}^{ + } ,n}} c^{{1 - \beta }}_{{{\text{Li}}^{ + } }} k^{\beta }_{{\text{f}}} {\text{R}}_{0} M_{{\text{S}}} }} {{{\text{D}}_{{{\text{LiS,S}}}} \rho ^{0}_{{\text{S}}} }}{\text{.}} $$

(163)