Formulation of a Statistical Mechanical Theory To Understand the Li

Article pubs.acs.org/JPCC

Formulation of a Statistical Mechanical Theory To Understand the Li Ion Conduction in Crystalline Electrolytes: A Case Study on Li-Stuffed Garnets Reginald Paul* and Venkataraman Thangadurai† Department of Chemistry, University of Calgary 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada

ABSTRACT: Ionic conductivity in solids is being computed using a wide range of computational methods such as molecular dynamics simulations and is measured using experimental methods, including electrochemical impedance spectroscopy and dc methods, and solid-state nuclear magnetic resonance spectroscopy. We report for the first time a statistical mechanical approach to estimate Li ion conductivity in the crystalline Li-stuffed garnet-type structure Li5La3Ta2O12, Li5.5La2.5Ba0.5Ta2O12, and Li6La2BaTa2O12. The estimated conductivity and activation energy for ionic conduction were found to be very consistent with experimental values for all three investigated garnets. The ionic conductivity was computed from the electrostatic friction coefficient of the Li ion using a combination of nonequilibrium statistical mechanics and electrostatics. The developed theory is derived from the fundamental transport equations that can be adapted to a wide range of crystalline ceramics electrolytes where crystallographic information is available, and sophisticated computational software and equipment may not be needed. degrees of freedom. Within the first category, Kumar and Yashonath9 have investigated the sodium ion conduction in NASICON, Na1+xZr2SixP3−xO12 (0 ≤ x ≤ 3), at a temperature of 600 K. These authors have used an empirical potential equation that is a linear combination of r−1, r−n, and r−6 with n ≠ 1 or 6. They are able to calculate Na ion conductivity of NASICON that is in excellent agreement with experiment and can predict the sodium ion pathways. Using the GULP suite of programs and potentials that are consistent with the crystal geometry, Stokes and Islam10 have studied the effects of defects and proton-dopant association in perovskite-type BaZrO3 and BaPrO3 with a favorable appearance of p-type conduction. For the second category of MD applications, the work of Xu et.al11 is of particular significance to this work. Using, density functional theory (DFT), Liivat and Thomas12 investigated the lithium migration pathways in Li2FeSiO4 cathode materials. In addition to the above papers that involve the numerical integration of Newton’s laws, there have been studies of the transport process based on hopping mechanism involving Monte Carlo simulation techniques. For example, Knödler, Pendzig, and Dieterich,13,14 considered the diffusion of charge carriers in a solid medium with an energy landscape characterized by randomly distributed immobile charge centers.

1. INTRODUCTION The last two decades have been a period of time in which both the scientific community and society at large have become deeply concerned about the negative impacts of fossil fuels and a quest for alternative energy sources such as electrical sources has become the focus of intensive studies. As a consequence, it is natural that materials with high electrical conductivity are much sought after. Already, materials such as membrane electrolytes, for example, Nafion, have emerged. However, in regions of higher temperatures solid inorganic substances are much more suitable, and a vast variety of such compounds have been synthesized and subjected to rigorous experimental and theoretical analysis.1−7 This paper is an attempt to provide a statistical mechanical formula for calculating the conductivity of crystalline solid electrolytes. However, it must be borne in mind that a vast plethora of theoretical work already exists. It is impossible for us to review this literature, and we will restrict ourselves to examples that have a relatively close affinity to our work. The fundamental principle underlying our work consists of selecting a suitable reaction coordinate that portrays the progress of a charge-bearing ion through the crystalline lattice and using the Liouville equation of motion8 for the distribution function. Because Newton’s laws of motion constitute the characteristic equations of Liouville equation, it is natural to consider molecular dynamics (MD) simulations as belonging to the extended family of our work. In reviewing the relevant literature on MD simulations, it is convenient to divide them into two classes: (i) studies that introduce interaction potentials as input with empirical parameters and (ii) studies in which the interaction parameters are calculated from the quantum mechanics of the electronic © XXXX American Chemical Society

2. THEORETICAL ASPECTS 2.1. Friction Coefficient of the Lithium Ion. During recent years, a considerable amount of information regarding Received: June 14, 2017 Revised: July 19, 2017

A

DOI: 10.1021/acs.jpcc.7b05837 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C the possible pathways that a lithium ion can take during its transport in garnet crystals has become available through the work of Thangadurai and coworkers.15 This work is based on a combination of crystallography and an empirical theoretical technique referred to as the bond valence (BV) method. Extensive literature and reviews on the BV approach have now become available;16 however, we will only focus our attention on some aspects of this work that impinge on our studies. Basically, we consider a Li+ ion located by a vector rLi that is bonded, in its first coordination layer, to a set N of counterions X. The bond valence sum (BVS) is defined in terms of the bond valence sLi,X (rLi, |rLi − rb|); b ∈ N as

Figure 2. Idealized diagram showing geometric and charge distribution details of a path element used in this model.

N

V (rLi) =

∑ sLi,X (rLi, |rLi − rb|) b=1

(1)

analytically, leaving a highly convergent sum, the leading term of which provides an excellent approximation (GronbechJensen, Hummer, and Beardmore,18 Jackson,19 and Paddison, Paul, and Zawodzinski20). The final ion-lattice potential is given by the following expression

Here the function V (rLi) (hence sLi,X (rLi, |rLi − rb|)) is required to be such that when the position rLi of the ion Li+ is an equilibrium position in the crystal then it (V (rLi)) must be equal to the absolute value of the oxidation state (Videal) of Li+. It follows that there are equilibrium points rLi,e where the mismatch defined as ΔV (rLi) ≡ |V (rLi) − Videal| must vanish. There will also exist points rLi,c that are close to equilibrium where the mismatch is small but not zero. To consider these near-equilibrium points in a quantitative manner, a small arbitrary cutoff or threshold values ΔVc of ΔV was selected such that any point rLi that possesses a mismatch ΔV ≤ ΔVc will be considered to be close to equilibrium. It has been postulated that the lithium ion pathway is the locus of points rLi,c.15,17 Adams and Rao17 define the pathways as follows: “These pathways are visualized as regions enclosed by isosurfaces, ...” which in this case are the points that are enclosed in surfaces where ΔV = ΔVc. Within each pathway, the lithium ion interacts with the other ions and, in particular, the negatively charged oxygen ions, which constitute their first coordination layer. Furthermore, it is evident from this work that the lithium ion passes through periodic sequences of octahedral and tetrahedral arrangement of oxygen ions experiencing electrostatic friction. It is our objective to compute this friction by modeling the pathways as a sequence of periodically charged cylinders of length l and radius R (Figures 1 and 2). The summation of the Coulombic potentials, causing the friction in such an arrangement, can be partially carried out

⎛ 2π ⎞ ⎛ 2πz ⎞ ⎟ U (r) ≡ U (ρ , z) = Ψ0K 0⎜ ρ⎟ cos⎜ ⎝ l ⎠ ⎝ l ⎠

(2)

Here Ψ0 is a constant, K0 is the modified Bessel function of the second kind, while ρ and z are, respectively, the radial and axial cylindrical coordinates with the limits 0 ≤ z ≤ l and σ′ ≤ ρ ≤ R (where σ′ is the hard sphere radius of the lithium ion). It is important to emphasize that in a dense solid of the type being studied here the van der Waals interactions must play an important role and will be included in future work; however, for the present we have not considered them. This should not be interpreted as a minimization of their importance. Their inclusion will, no doubt, alter the situation and, as mentioned, will be explored further. The frictional force, in the present context, is the reactive force exerted by the lattice on the lithium ion due to the motion of the latter. It is convenient to fix the coordinate origin on the lithium ion, which will consequently appear stationary but the lattice will appear to move with a velocity of −v. For the momentum p of the lattice, we consider it to be a particle of mass m, which is several orders of magnitude greater than the mass of the lithium ion. The Hamiltonian and the corresponding Liouville operator (McQuarrie8) will be given by H(pz , pρ , z , ρ) =

pz2 2m

+

pρ2

⎛ 2π ⎞ ⎛ 2πz ⎞ ⎟ + Ψ0K 0⎜ ρ⎟ cos⎜ ⎝ l ⎠ ⎝ l ⎠ 2m (3)

⎡p pρ ∂ 2π Ψ0 ⎛ 2πz ⎞ ⎛ 2π ⎞ ∂ ∂ ⎟K ⎜ ρ⎟ L(z , ρ , pz , pρ ) = − i⎢ z sin⎜ + + ⎝ l ⎠ 0⎝ l ⎠ ∂p ⎢⎣ m ∂z m ∂ρ l z +

⎤ 2π Ψ0 ⎛ 2πz ⎞ ⎛ 2π ⎞ ∂ ⎥ ⎟K ⎜ cos⎜ ρ⎟ 1⎝ ⎝ ⎠ ⎠ l l l ∂pρ ⎥⎦

(4)

Here pz and pρ are the components of p along the z and ρ axes of the cylindrical frame, respectively. Using the standard techniques of statistical mechanics, the force components experienced by the lattice due to the lithium ion will be given by Fz = iLpz and Fρ = iLpρ, respectively. If the external field acts along the z axis, then we consider Fz only

Figure 1. Schematic diagram showing the Li ion pathways in Li-stuffed garnet as a set of covering cylinders drawn on the BVS iso-surfaces and used to develop electrostatic friction coefficient of the Li ion using a combination of nonequilibrium statistical mechanics and electrostatics model. The BVS iso-surfaces were adopted from ref 15.

Fz(z , ρ) = iLpz = B

2π Ψ0 ⎛ 2πρ ⎞ ⎛ 2πz ⎞ ⎟ sin⎜ ⎟ K 0⎜ ⎝ l ⎠ ⎝ l ⎠ l

(5)

DOI: 10.1021/acs.jpcc.7b05837 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Here Fz(z, ρ) is the force at an initial point in time, which we take to be t = 0, and the value of this force at a later point in time t will be given by Fz(z, ρ, t) = eiLtFz(z, ρ). This leads to the force−force correlation function

c 2 = 4π 2kT

∞

∞

∫−∞ dpz ∫−∞ dpρ ∫0

=

l

dz

∫σ

R

∫0

∞

(6)

1 dt ⟨FzFz(t )⟩ = kT

∫0

d t G (t )

= (1/Q ) exp[−H(pz , pρ , z , ρ)/kT ] ∞

∞

l

dz

∫σ

∫0

dz

∫σ

G (t ) ≈ (8)

dρ

ζ=

σ=

2.2. Calculation of the Force−Force Correlation Function. The force−force correlation function can be written as a power series, that is

n=0 ∞

=

∑ n=0

n!

⟨Fz(iL)n Fz⟩

( −t )n + 1 cn + 2 n!

D= (9)

ϕ

|α | 1+

|β | 2 t 2 |α |

(14)

1 kT

∫0

∞

⎡ ⎛ | β | 2 ⎞⎤ π α 3/2 d t ⎢| α | / ⎜1 + t ⎟⎥ = ⎣ ⎝ 2|α| ⎠⎦ kT 2β

(15)

lqLi2ρLi 2π 2 Ψ0 mc I13/2 (16)

kT ζ

(17)

Electrical resistance of the garnet crystal to the lithium ion current can be estimated using conductivity, that is

In keeping with standard terminology, we will refer to the quantities cv introduced above as the vth moment defined as follows cv = ⟨pz (iL)v pz ⟩

ϕ

ϕ

2

Here ρLi is the number density of the lithium ions in the garnet crystal. Diffusion coefficient of the lithium ion can be computed using the relationship

G(t ) = ⟨Fz e Fz⟩

∑ (−t )

⎛ 2πz ⎞ ⎛ 2πρ ⎞ ⎟K ⎜ ⎟ sin 2⎜ ⎝ l ⎠ 1⎝ l ⎠

3kT (I2 − I1) + 2Ψ0[I3 − 3(I4 + I5) + I6]

iLt

=

, I6 =

ϕ

With the friction coefficient given by eq 15, the conductivity can be calculated by using standard formulas to yield

dρ exp[U (ρ , z)/kT ]

n

⎛ 2πz ⎞ ⎛ 2πρ ⎞ ⎟K ⎜ ⎟ cos2⎜ ⎝ l ⎠ 1⎝ l ⎠

ϕ 2

The physical quantity that we are interested in is the friction coefficient defined in eq 7, which yields

R

R

∞

, I4 =

ϕ

(13)

= (2πmkT )ς

ς=

I5 =

⎛ 2πz ⎞ ⎛ 2πρ ⎞ ⎟K ⎜ ⎟ sin 2⎜ ⎝ l ⎠ 0⎝ l ⎠

2

⎛ 2πz ⎞ ⎛ 2πρ ⎞ ⎟K ⎜ ⎟ cos⎜ ⎝ l ⎠ 2⎝ l ⎠

Here the subscript ϕ implies that only the spatial part of the distribution function as defined in eq 8 is involved. With these definitions for the moments we have, using the shorthand symbols −c2 = |α| and c4 = |β|, for the correlation function G and in terms of the [0, 2] Padé approximant21

(7)

exp[−H(pz , pρ , z , ρ)/kT ]

l

I3 =

⎛ 2πz ⎞ ⎛ 2πρ ⎞ ⎟K ⎜ ⎟ cos2⎜ ⎝ l ⎠ 0⎝ l ⎠

2

, I2 =

∞

feq (z , ρ , pz , pρ ) ≡ ψ (pz )ψ (pρ )ϕ(z , ρ)

∫−∞ dpz ∫−∞ dpρ ∫0

⎛ 2πz ⎞ ⎛ 2πρ ⎞ ⎟K ⎜ ⎟ cos⎜ ⎝ l ⎠ 0⎝ l ⎠

I1 =

Here the function feq is the equilibrium distribution function and the pointed brackets are the equilibrium statistical mechanical averages. The relevant definitions are as follows

Q=

(11)

The quantities Ij are defined as follows

From Newton’s third law, we realize that the medium will, by reaction, exert the same force back on the lithium ion, and thus G(t) will also be the force−force correlation function for the lithium ion. Again, invoking the principles of nonequilibrium statistical mechanics, the friction coefficient of the lithium ion will be given by a time integral of the correlation function 1 ζ= kT

I1

(12)

dρ Fz(z , ρ)eiLt Fz(z , ρ)

feq (z , ρ , pz , pρ )

l2

8π 4kT Ψ0[3kT (I2 − I1) + 2Ψ0{I3 − 3(I4 + I5) + I6}] l 4mc

c4 =

G(t ) = ⟨Fz eiLt Fz⟩

Ψ0

R=

(10)

⎛L⎞1 ⎜ ⎟ ⎝ A⎠σ

(18)

Here L is the length of the crystal along the z axis and A is the area of cross section of the crystal normal to the z axis.

To use eq 9 to compute G(t), it would appear, at first, that all infinite moments would have to be calculated; this is clearly impossible and not necessary. It is sufficient to calculate a few of the leading terms, which in the present case are c2 and c4, and then approximate the remainder by using the method of Padé approximants. It is important to bear in mind that in the present classical framework only even moments (v) are nonvanishing. The calculation of the moments is fairly straightforward but tedious, and hence only the final results will be given

3. RESULTS AND DISCUSSION In this study, our interest has been focused on the garnet-type compounds possessing a general formula unit Li5+2xLa3−xBaxTa2O12 (x = 0, 0.5, and 1), for which much experimental data is available on both the crystal structure and from electrical measurements.22 We have commenced our studies from the simplest of these with x = 0 member, Li5La3Ta2O12. Crystal structure studies have shown that this C

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The Journal of Physical Chemistry C Table 1. Computational and Experimental Conductivity of Li-Stuffed Garnet-type Li5La3Ta2O12 T (K)

296

322

345

386

Ψ0 (J) DTh (m2/s) RTh (Ω) σTh (S/m) Rexp. (Ω) σexp. (S/m)

−3.54 × 10−18 2.1 × 10−15 127639 2.49 × 10−4 128087 2.48 × 10−4

−1.33 × 10−18 1.83 × 10−14 15871.5 2.0 × 10−3 15872.5 2.0 × 10−3

−5.77 × 10−19 1.15 × 10−13 2689.8 1.10 × 10−2 2676.2 1.18 × 10−2

−1.88 × 10−19 1.45 × 10−12 240.1 0.13 240.6 0.13

Table 2. Computational and Experimental Conductivity of Li-Stuffed Garnet-type Li5.5Ba0.5La2.5Ta2O12 T (K) Ψ0 (J) DTh (m2/s) RTh (Ω) σTh (S/m) Rexp. (Ω) σexp. (S/m)

296

325 −19

−7.9 × 10 3.62 × 10−14 6275.3 5.07 × 10−3 6332.1 5.02 × 10−3

345 −18

−4.49 × 10 1.42 × 10−13 1761.3 1.80 × 10−2 1768.4 1.80 × 10−2

393 −19

−2.47 × 10 5.62 × 10−13 480.8 6.61 × 10−2 481.4 6.61 × 10−2

−1.15 × 10−19 3.51 × 10−12 86.8 0.37 86.9 0.37

Table 3. Computational and Experimental Conductivity Of Li-Stuffed Garnet-type Li6BaLa2Ta2O12 T (K)

296

325

349

393

Ψ0 (J) DTh (m2/s) RTh (Ω) σTh (S/m) Rexp. (Ω) σexp. (S/m)

−7.9 × 10−19 3.62 × 10−14 6275.3 5.0 × 10−3 6332.1 5.0 × 10−3

−4.49 × 10−19 1.42 × 10−13 1761.3 1.8 × 10−2 1768.4 1.8 × 10−2

−2.47 × 10−19 5.62 × 10−13 480.8 6.61 × 10−2 481.4 6.61 × 10−2

−1.15 × 10−19 3.51 × 10−12 86.8 0.37 86.8 0.37

crystal belongs to the space group Ia3d̅ , which is, of course, the well-established space group for the traditional garnets A3IIB2III(SiO4)3. If we consider just the ionic component [La3Ta2Li3O12]2−, then the structural analogy with the traditional garnets would be exact, in which the lanthanum, tantalum, and the lithium ions are located in the Wyckoff positions 24c, 16a and 24d, respectively. Furthermore, the oxygen ions are in the general Wyckoff position 96h, where they provide a cubic, octahedral, and tetrahedral environment for the lanthanum, tantalum, and the lithium ions, respectively.23,24 In the case of Li5La3Ta2O12, two extra lithium ions must be accommodated, which are forced to occupy the relatively higher energy octahedral sites in the Wyckoff position 16b; the enhanced conductivity of the crystal is then attributed to these mobile lithium ions. In keeping with this hypothesis, it was conjectured that if the number of lithium ions could be further increased so that some of them are driven into the general Wyckoff positions, then the conductivity of the crystal could be further increased. This has resulted in the synthesis of what have been referred to as the lithium “stuffed” garnets and measurements of their conductivity. The latter have, indeed, vindicated the belief that conductivity will be enhanced. There are, in fact, two effects on the conductivity that need to be explained based on a suitable theoretical model: (1) increase in conductivity with temperature and (2) increase in conductivity observed with elevated lithium content (the “stuffed” garnets). At first sight, both of these observations could be explained as the result of increased charge-carrier density in the crystal and the elevated average speed of the carriers with temperature. However, we show here that there is, in addition, a cooperative effect that arises from changes in the electrostatic friction given by eq 15 through the surface potential (Ψ0). Tables 1−3 display the effects of changes in Ψ0 as the temperature is varied and the affects of the changes in the

lithium ion density on the following: diffusion coefficients, resistances, and conductivities of Li5La3Ta2O12 with 40 Li+ per unit cell and lattice parameter of 12.81 Å, Li6BaLa2Ta2O12 with 48 Li+ per unit cell and lattice parameter of 12.946 Å, and Li5.5Ba0.5La2.5Ta2O12 with 44 Li+ per unit cell and lattice parameter of 12.813 Å. The values in the columns that are headed by σTh were calculated from eq 16. Experimental evidence points to the fact that there is a substantial increase in the conductivity with temperature. This is, of course, not surprising because a temperature increase would result in increased kinetic energy of the lithium ions and consequently their speed of transport. However, the present theoretical calculation shows that temperature alone, while predicting a rise in conductivity, does not show a level of enhancement comparable to that experimentally observed unless the potential parameter Ψ0 is also varied. This suggests that there is a cooperative effect involving both T and Ψ0. In Figure 3 we show the insufficient rise in conductivity with temperature alone for a fixed value of Ψ0 = −3.54 × 10−18 J for Li5La3Ta2O12. It is possible to draw a similar graph in which the temperature is held fixed and the parameter Ψ0 is varied. However, Ψ0 is not an experimentally controlled quantity, and such a diagram would not be very illuminating. It is more useful to present an Arrhenius plot in which both Ψ0 and T are changed to demonstrate the power of σ as given by eq 16 to reproduce the experiment (Figure 3). These plots show that there are synergies between the effects of temperature and Ψ0, or, in other words, temperature changes alter the lattice so that Ψ0 is also altered. It is useful to present a qualitative reasoning behind this observation. As the temperature is raised the average speed of the lithium ions increases, which causes a partial rise in conductivity. However, the oxygen ions O2−, which reside in the general Wyckoff position 96h, are also susceptible to increase in kinetic energy, and, furthermore, D

DOI: 10.1021/acs.jpcc.7b05837 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

which is carrying the current and the oxygen ions that are responsible for the friction. The arrival of other lithium ions in the vicinity of the oxygen will result in shunting of some of the Faraday lines away from the ion carrying the current and thus reducing Ψ0. We suggest that further theoretical confirmation regarding the change in conductivity with temperature and the resulting variation of Ψ0 may be obtained by carrying out two MD calculations: (1) with a rigid lattice (except for the conducting Li+ ions) and (2) with a flexible lattice. If the conjectures presented here are correct, then the former calculation should yield poor agreement and the latter good agreement with experiment with regards to the variation of conductivity with temperature.

4. CONCLUSIONS In summary, we have derived, by using a combination of statistical mechanics and electrostatics, a relatively simple compact formula to estimate the ionic conductivity of the crystalline Li-stuffed garnet-type compounds Li5+2xLa3−xBaxTa2O12 (x = 0, 0.5, and 1). The results from this theory are found to be consistent with experiment. Furthermore, the increase in electrical conductivity of Li garnets as a result of increasing the lithium content and the temperature is attributed to a fall in the surface potential Ψ0 and a concomitant fall in the electrostatic friction coefficient. The formula can also be extended to other crystalline ionic conductors, for which crystallographic data are known. It is also important to mention that in this work we have primarily concentrated on the role of electrostatic friction; however, in the high-density solid being studied the van der Waals interactions also play a cardinal role, and their inclusion will quite likely alter the results in some way, which will be explored further.

Figure 3. Arrhenius plots showing a comparison of experimental and theoretical Li ion conductivity of Li-stuffed garnets. The Li ion conductivity was computed using the expression 16. A close overlap with experimental value22 was obtained using variable surface potential as a function of temperature.

because they carry negative charge they will move in a direction opposite to the lithium ions under the influence of the applied external field. The effects of this can be appreciated by reference to Figure 3, from which it can be concluded that there will be a fall in the negative charge in the vicinity of the transport path, resulting in a diminution of the value of Ψ0 and thus of the electrostatic friction (eq 16). From eq 16 the origins of both the effects of temperature and Ψ0 can be discussed. Temperature appears in the six thermal averages of the various order derivatives of the potential listed in eq 13. Unfortunately, the relevant integrations cannot be carried out analytically; however, a very simple numerical analysis shows that all six of these integrals can be very accurately written for a fixed value of Ψ0 as eq 19

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AUTHOR INFORMATION

Corresponding Author

a1(n) (19) T (n) (n) Here, a0 and a1 are temperature-independent. Furthermore, In(T), n = 3, 4, 5, 6, are much smaller than I1(T) and I2(T); therefore, due to the presence of the linear term, T, in the first term under the square root, the entire quantity with the square root is almost temperature-independent. The temperature rise is due to the fall in I1 appearing in the denominator with T,, and it is the fourth-order derivative of the potential in the direction of the external driving field. A similar analysis shows that these integrals fall linearly with Ψ0: In = bn0 + bn1Ψ0 As it is pointed out above that even in the case of Li5La3Ta2O12, which has the smallest number of stuffed ions, a substantial fraction of these ions is forced to occupy Wyckoff positions that are unable to provide the relatively stable tetrahedral coordination. Consequently, these ions will tend to approach the negatively charged surfaces enclosing the pathways of the current carrying ions and thus lowering the value of Ψ0. Physically, the meaning of this can be understood with the aid of the bond valence model, as discussed by Brown,16 according to which, “A bond is then defined as occurring between a cation and an anion if and only if they are directly linked by Faraday lines. The number of such lines is proportional to the electrostatic flux, which can be used as a direct measure of the strength of the bond.” In the present case, we are considering the bond that exists between a lithium ion, In(T ) = a0(n) +

*E-mail: [email protected]. ORCID

Venkataraman Thangadurai: 0000-0001-6256-6307 Author Contributions

R.P. developed the transport theory and wrote the paper. V.T. provided the experimental results and helped with formulation of the theory and commented on the manuscript. Notes

The authors declare no competing financial interest. † V.T.: E-mail: [email protected].

■

ACKNOWLEDGMENTS R.P. thanks the Department of Chemistry and University of Calgary for Emeritus professorship. The Natural Sciences and Engineering Research Council of Canada (NSERC) has supported this work through discovery grants to V.T. (award number: RGPIN-2016-03853).

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REFERENCES

(1) Shitara, K.; Moriasa, T.; Sumitani, A.; Seko, A.; Hayashi, H.; Koyama, Y.; Huang, R.; Han, D.; Moriwake, H.; Tanaka, I. FirstPrinciples Selection of Solute Elements for Er-Stabilized Bi2O3 OxideIon Conductor with Improved Long-Term Stability at Moderate Temperatures. Chem. Mater. 2017, 29 (8), 3763−3768. E

DOI: 10.1021/acs.jpcc.7b05837 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C (2) Wang, Y.; Richards, W. D.; Ong, S. P.; Miara, L. J.; Kim, J. C.; Mo, Y.; Ceder, G. Design Principles For Solid-State Lithium Superionic Conductors. Nat. Mater. 2015, 14 (10), 1026−1031. (3) Kuhn, A.; Duppel, V.; Lotsch, B. V. Tetragonal Li10GeP2S12 And Li7GePS8 - Exploring the Li-Ion Dynamics in LGPS Li Electrolytes. Energy Environ. Sci. 2013, 6 (12), 3548−3552. (4) Emly, A.; Kioupakis, E.; Van der Ven, A. Phase Stability and Transport Mechanisms in Antiperovskite Li3OCl and Li3OBr Superionic Conductors. Chem. Mater. 2013, 25 (23), 4663−4670. (5) Knauth, P. Inorganic Solid Li-Ion Conductors: An Overview. Solid State Ionics 2009, 180 (14−16), 911−916. (6) Saiful Islam, M.; Slater, P. R. Solid-State Materials for Clean Energy: Insights from Atomic-Scale Modeling. MRS Bull. 2009, 34 (12), 935−941. (7) Malavasi, L.; Fisher, C. A. J.; Islam, M. S. Oxide-Ion and Proton Conducting Electrolyte Materials for Clean Energy Applications: Structural and Mechanistic Features. Chem. Soc. Rev. 2010, 39 (11), 4370−4387. (8) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976. (9) Kumar, P. P.; Yashonath, S. Structure, Conductivity, and Ionic Motion in Na1+xZr2SixP3‑xO12: A Simulation Study. J. Phys. Chem. B 2002, 106 (28), 7081−7089. (10) Stokes, S. J.; Islam, M. S. Defect Chemistry and Proton-Dopant Association in BaZrO3 and BaPrO3. J. Mater. Chem. 2010, 20 (30), 6258−6264. (11) Xu, M.; Park, M. S.; Lee, J. M.; Kim, T. Y.; Park, Y. S.; Ma, E. Mechanisms of Li+ Transport in Garnet-Type Cubic Li3+xLa3M2O12 (M = Te, Nb, Zr). Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85 (5), 052301. (12) Liivat, A.; Thomas, J. O. Li-Ion Migration in Li2FeSiO4-Related Cathode Materials: A DFT study. Solid State Ionics 2011, 192 (1), 58− 64. (13) Knödler, D.; Pendzig, P.; Dieterich, W. Transport and Ac Response in a Model of Glassy Electrolytes. Solid State Ionics 1994, 70, 356−361. (14) Knödler, D.; Pendzig, P.; Dieterich, W. Ion Dynamics in Structurally Disordered Materials: Effects of Random Coulombic Traps. Solid State Ionics 1996, 86, 29−39. (15) Thangadurai, V.; Adams, S.; Weppner, W. Crystal Structure Revision and Identification of Li+-Ion Migration Pathways in the Garnet-Like Li5La3M2O12 (M = Nb, Ta) Oxides. Chem. Mater. 2004, 16 (16), 2998−3006. (16) Brown, I. D. Recent Developments in the Methods and Applications of the Bond Valence Model. Chem. Rev. 2009, 109 (12), 6858−6919. (17) Adams, S.; Rao, R. P. Modelling of Ion Transport in Solids with a General Bond Valence Based Force-Field. At. Indones. 2011, 36, 95− 104. (18) Gronbech-Jensen, N.; Hummer, G.; Beardmore, K. M. Lekner Summation of Coulomb Interactions in Partially Periodic Systems. Mol. Phys. 1997, 92 (5), 941−946. (19) Jackson, J. D. Classical Electrodynamics; John Wiley & Sons Inc., 1975. (20) Paddison, S. J.; Paul, R.; Zawodzinski, T. A. A Statistical Mechanical Model of Proton and Water Transport in a Proton Exchange Membrane. J. Electrochem. Soc. 2000, 147 (2), 617−626. (21) Paul, R. Applications of Padé Approximants to Correlation Functions. Mol. Phys. 1980, 41 (1), 1−17. (22) Truong, L.; Colter, J.; Thangadurai, V. Chemical Stability of LiStuffed Garnet-Type Li5+xBaxLa3−xTa2O12 (x = 0, 0.5, 1) in Water: A Comparative Analysis with the Nb Analogue. Solid State Ionics 2013, 247−248, 1−7. (23) Cussen, E. J. The Structure of Lithium Garnets: Cation Disorder and Clustering in a New Family of Fast Li+ Conductors. Chem. Commun. 2006, 4, 412−413. (24) O’Callaghan, M. P.; Cussen, E. J. Lithium Dimer Formation in the Li-Conducting Garnets Li5+XBaxLa3‑XTa2O12 (0 < X ≤ 1.6). Chem. Commun. 2007, 20, 2048−2050. F

DOI: 10.1021/acs.jpcc.7b05837 J. Phys. Chem. C XXXX, XXX, XXX−XXX