Transport of secondary carriers in a solid lithium-ion conductor

5 Inorganic-solid lithium electrolytes are typically thought of as single-ion conductors, but the presence of secondary carriers can strongly aﬀect their electrical responses. Conventional descriptions of multi-carrier transport neglect both interactions between mobile species and stress diﬀusion — phenomena which can markedly impact the electrical response. We apply irreversible thermodynamics to develop a chemomechan-ical transport model for elastic-solid ionic conductors containing two mobile ions. We simulate lithium-ion conducting Li 5 La 3 Nb 2 O 12 (LLNO) garnet oxide, a material within which experiments have shown that mobile protons can be freely substituted for lithium to form Li 5(1 − y ) H 5 y La 3 Nb 2 O 12 . When subjected to a current, we ﬁnd that proton-substituted LLNO exhibits bulk lithium polarization, whose extent is partially controlled by cation/cation interactions. Secondary carriers segregate naturally if their global concentration is low, accumulating in a thin boundary layer near the cathode. We quantify the limiting current and Sand’s time, and analyze experimental data to show how competitive proton transport aﬀects LLNO performance. More conductive, durable electrolyte materials would enable higher-power, safer energy storage systems. 6 Solid electrolytes have promising mechanical properties, but are generally less conductive than liquids and 7 can be prone to chemical instability [1] and dendrites [2]. In solid lithium-ion conductors, dopants have 8 exploited enhance ionic conductivity and chemical impurities and defects 9 observed to facilitate material degradation.

Locally, the excess charge density produces an external electric field E, governed by Poisson's equation where is the electrolyte's dielectric permittivity, taken to be constant. Species continuity is expressed by for k = 1, ..., n. A momentum balance relates system dynamics to the external mechanical stress σ and electrostatic body forces ρ e E, Goyal and Monroe suggest that an additional diffusion stress can appear here because species in multicom-56 ponent materials convect momentum at their own velocities, rather than the mass-average velocity [10]. 57 Diffusion stress is generally small and has been neglected. 58 Hirschfelder et al. applied principles of irreversible thermodynamics to show how flux laws in a multi-59 species system can be derived from an expression for local energy dissipation [11]. Given a statement of 60 the dissipation function, one can identify conjugate pairs of species velocities and thermodynamic driving 61 forces d k that contribute to the energy losses associated with diffusion. OSM equations that describe how 62 the driving forces balance the diffusional drag produced by the relative motion of species take the form 63 in which K kj is a coefficient that quantifies diffusional drag between species k and j. The dissipation function 64 derived by Goyal and Monroe [10] shows that d k breaks down into a mass-diffusion force, arising from the electrochemical potential gradient of species k, and a stress-diffusion force, from mechanical gradients. 66 Expressing this force per mole of species k gives 67 d k = − ∇µ k + M k ρ ∇p + : ∇ τ .
Here is the deformation-strain tensor, p the external pressure, and τ = σ − p I is the deformation stress.

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Because the Gibbs-Duhem relation requires k c k d k = 0, one only needs to write constitutive laws for n − 1 69 driving forces in an n-ary diffusion system.
where sites in the crystal lattice are designated as species m. Here R is the gas constant and T is the absolute 73 temperature; χ kj is a thermodynamic factor, which describes how the logarithm of species k's activity varies 74 with species j's lattice occupancy ξ j , and V k is the partial molar volume of species k. 1 The system of Eqs. 1 75 to 6 is closed by defining the lattice occupancy and stating constitutive laws for the thermodynamic material 76 properties χ kj , V k , and , as well as the transport properties K kj .

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Constitutive laws. We posit that the L 1−y H y LNO material can be treated as a ternary system (n = 3), in write independent OSM laws for the forces d + and d p .

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A Stefan-Maxwell form, which introduces the diffusivity of species k through species j, D kj , is appropriate 84 for the drag coefficients: Onsager reciprocity guarantees that K kj = K jk , so D kj = D jk . Following the approach suggested by 86 Fornasiero et al. [14], Eq. 7 has been modified from the form typically used for liquids, in that the lattice-87 site concentration c − , rather than the total concentration of all species, appears in the denominator. This 88 choice can alternatively be justified by a formal assumption that species moving in conductive channels have 89 very small partial molar volumes compared to the lattice, V + , V p << V − , so that the total molar volume 90 of the electrolyte depends very weakly on its lattice occupancy.

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The Monroe-Delacourt process [12] converts Eqs. 4 and 5 into a modified form of Ohm's law, and a flux-explicit transport law for J − p = c p ( v p − v − ), the excess molar flux of protons relative to the 93 lattice, As well as the new parameter which expresses the ratio of the secondary carrier's charge to that of lithium, Eqs. 8 and 9 introduce 96 three experimentally observable transport parameters: the ionic conductivity κ, the lithium transference  1 Since extensivity of volume c k V k = 1, the specification of partial molar volumes makes cn a dependent variable, justifying the removal of the n th term from the sum in equation 6. ξ + = c + /c − and ξ p = c p /c − for Li + and H + , respectively. We also introduce the symbol ξ 0 represent the equilibrium lattice occupancy of carriers in the single-ion conductor limit (i.e., the equilibrium 103 Li + occupancy of proton-free LLNO).

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Some alternative parameter definitions help to express the macroscopic properties in more tangible terms. 105 We first replace the lithium/lattice Stefan-Maxwell diffusivity D +− with which represents the electrolyte's equilibrium ionic conductivity in the absence of the secondary carrier.

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That is, κ 0 is the electroneutral bulk conductivity of LLNO (L 1−0 H 0 LNO, in which c p = 0 uniformly). Two 108 additional convenience parameters, express the diffusional resistances that protons place on the negative lattice and lithium ions, respectively, 110 per unit of resistance the lattice exerts on lithium ions.

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By manipulating the equations above, one can express all three macroscopic properties as functions of 112 the local carrier occupancies: and These functions of ξ + and ξ p are parameterized by the properties κ 0 , c − , ξ 0 + , ζ p+ , r p+ , and r p− .

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For simplicity we describe L 1−y H y LNO chemically as an ideal solid solution. In such a material the 117 configurational entropy is the only composition-dependent part of the total entropy. A lattice-gas model 118 can be used to establish the configurational contribution to the free energy, yielding a set of independent 119 thermodynamic factors summarized as which has been defined so that it reduces to the identity matrix in the limit that ξ p + ξ + << 1.

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Finally, we assume that all L 1−y H y LNO materials are isotropically linear-elastic [15], so that the mechanical state is determined by Geometry and boundary conditions. Simulations will be performed to describe planar L 1−y H y LNO slabs of 131 thickness L, sandwiched between two planar electrodes -spatially one-dimensional systems. As argued 132 earlier [9,17], deformation stresses vanish in this configuration, and the normal stress in the x direction 133 relates simply to pressure, through σ xx = 3p − 2p θ . One can also combine Eqs. 1 and 3, then integrate to 134 produce an equation for local pressure in terms of E, the x-component of the electric field, This result neglects inertial contributions to momentum, which are very small at practical current densities.  also imply that the mass-average velocity in the x-direction within the electrolyte there is v = iM + /(ρF z + ).

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Following our earlier approach [9], we incorporate an interfacial characteristic frequency (R int C int ) −1 145 as a model parameter to set the electric field at the boundaries: implying that the interface is very capacitive and there is a much steeper polarization near interfaces than 148 in the bulk. We use 2πR int C int = 10 −1 s when considering reactive interfaces here. LLNO is ξ 0 + = 5/9.

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The density of pure LLNO at 300 K and atmospheric pressure is ρ θ = 5.170 gcm −3 [23]. In line with the 162 assertion that carrier concentration minimally affects lattice dimensions, we let Hence the density measurement is consistent with a partial molar volume for sites of V restricts composition such that ξ + + ζ p+ ξ p = ξ 0 + . Thus, if the bulk conductivity is insensitive to y in 172 L 1−y H y LNO, then κ/κ 0 = 1 when ξ + = ξ 0 + − ζ p+ ξ p , independent of ξ p . Through Eq. 13, this requires 173 that r p− = ζ p+ . Since ζ p+ = 1 for L 1−y H y LNO, the invariance of ionic conductivity with respect to y 174 implies that Li + and H + have similar mobilities through the lattice: In electrolytes like L 1−y HyLNO, the partial molar volumes of both Li + and H + are extremely small relative to that of the negative sublattice. Ion-size effects depend on the absolute difference between the carriers' partial molar volumes and are therefore neglible here. When this difference is large, there can be considerable ion-size effects in the double layer, as has been shown for ionic-liquid-based lithium electrolytes [24].
Thus the conductivity of L 1−y H y LNO is proportional to total lattice occupancy, and its lithium transference This establishes an order-preserving, invertible map between the closed domain ψ ∈ [−1, 1] and the semi- of ion pairs, as has been suggested to explain ionomers that exhibit negative transference numbers [31]. presence of H + slows Li + transport, increasing bulk polarization relative to the non-interacting case; when 227 it is negative, the opposite is observed.

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As the space-charge layer adjacent to the negative electrode screens the electric field, it accumulates a form identified by inserting Eqs. 5, 6, and 16 into the diffusion driving force that appears in Eq. 9, then 246 using the electroneutrality constraint to eliminate ξ + . When ζ p+ = 1, as it does for L 1−y H y LNO, this can 247 be used to show that Eq. 23 becomes after insertion of Eqs. 13 through 15 and 22. 249 Under local electroneutrality the current density is always constant with respect to x, and at steady 250 state the proton flux vanishes uniformly. Thus Eq. 25 produces a simple differential equation in ξ p , wherein the rightmost equality introduces a dimensionless current density, I. Direct integration and use of 252 Eq. 19 to specify the unknown constant yields the steady-state proton distribution. This result also determines the lithium distribution because local 254 electroneutrality requires that ξ + (x) = ξ 0 + − ξ p (x).

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Expressed in dimensional terms, the limiting current density i L is   proton flux from Eq. 25 into the proton balance from Eq. 2, one arrives at the convective diffusion equation A spatial first derivative appears in Eq. 31 due to the composition dependence of lithium transference. Thus 316 the dimensionless current I is revealed to be formally analogous to a Péclet number, although its appearance 317 in the proton balance owes to migration (carrier drift), rather than convection.

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Eq. 31 was solved to simulate the response at times τ > 0 to a step change up to dimensionless current I, analytical results that work well at short times and also predict the proton content precisely at particular 327 locations.

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Relaxation to the steady state. Laplace transformation with respect to time changes the partial differential is phrased in terms of the error function complement and its first integral. 4 With sums truncated after the 339 fifth terms, Eq. 33 produces results that differ by less than 5 × 10 −11 everywhere from the steady-state parameter that also controls system relaxation. Ultimately, when analyzing asymptotic behavior of this 368 problem at a given instant, one also needs to consider how the value of parameter β, defined as compares to the inverse diffusive penetration depth 1/(2 √ τ ). Unlike what one would expect from typical diffusion systems, β Sand is not a universal constant; its variation 384 with y owes to the composition dependence of lithium transference.

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Since the dynamics of the proton distribution is determined by lithium/proton interactions, experimental of global proton content y as well as ψ, Usefully, the sample thickness does not appear here, although it is involved in the choice of applied current, Dendrite formation as a result of lithium depletion near the cathode could be a problem for less conduc-407 tive, multicarrier solid electrolytes operated at current densities in the range of 0. protons in a thermodynamically consistent way.

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The full non-electroneutral model was solved numerically to describe single-crystal L 1−y H y LNO materials