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Supplementary information

The activation entropy for ionic conduction and critical current density for Li charge transfer in novel garnet-type Li6.5La2.9A0.1Zr1.4Ta0.6O12 (A = Ca, Sr, Ba) solid electrolytes

Sanoop Palakkathodi Kammampata†, Hirotoshi Yamada‡, Tomoko Ito‡, Reginald Paul†, and Venkataraman Thangadurai†*

†Department of Chemistry, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4. ‡Graduate School of Engineering, Nagasaki University, 1-14, Bunkyo-machi, Nagasaki 852-8521, Japan

1. Derivation of friction coefficient of Li ion and relation with conductivity

The total Coulombic potential experienced by the Li ion within such a cylindrical element of the

pathway is given by:

(1)𝑈(𝜌,𝑧) = Ψ0𝐾0(

2𝜋𝑙

𝜌)cos (2𝜋𝑧𝑙 )

Here, is a constant, which depends on the charges of the Li ion and the counterions (in the Ψ0

present case oxygen ions). The distribution of the negatively charged oxygen ions in the form of

periodic octahedral and tetrahedral arrangements are assumed to result in periodic negatively

charged rings on the cylinder wall. In, addition, since we are dealing with charge-charge

interactions, also includes a dependence on the dielectric constant. The precise mathematical 0

form of appears in the geometrical considerations that are used in the Coulombic summation 0

(used in the derivation of Eq.(1)) procedure developed by Gronbech-Jensen, Hummer and

Beardmore.1 However, in this paper we treat as a parameter that is determined from experiment 0

hence it is not necessary to write down its detailed mathematical form. is the modified Bessel’s 0K

Electronic Supplementary Material (ESI) for Journal of Materials Chemistry A.This journal is © The Royal Society of Chemistry 2019

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functions of the second kind while and are the radial and axial cylindrical coordinates with z

the limits and (where is the hard sphere radius of the Li ion).0 z l 𝜎𝐻 ≤ 𝜌 ≤ 𝑅 𝜎𝐻

The frictional force, in the present context, is the reactive force exerted by the lattice on

the Li ion due to the motion of the latter. The Hamiltonian and the corresponding Liouville

operator (McQuarrie)2 are given by:

(2) 22

0 02 2, , , ( ) cos

2 2z

z

pp zH p p z Km m l l

(3)

0 00 1

, , ,

2 22 2 2 2sin cos

z

z

z

L z p p

pp z zi K Km z m l l l p l l l p

Here, and are the components of along the and axes of a cylindrical frame zp p p z

respectively.

Using the standard techniques, the force component experienced by the lattice due to the

Li ion in the direction will be given by . If the external field acts along the axis we z z zF iLp z

consider only:zF

(4) Fz z, iLpz

20

lK0

2l

sin 2 zl

Here, is the force at an initial point in time, which we take to be and the value of ,zF z 0t

this force at a later point in time will be given by: . This leads to the t , , ,iLtz zF z t e F z

force-force correlation function:

(5) iLtz zG t F e F

3

Invoking the principles of non-equilibrium statistical mechanics, the friction coefficient of the Li

ion will be given by a time integral of the correlation function (commonly referred to as the Kubo-

Green formula):

(6)𝜁 =

1𝑘𝑇

∞

∫0

𝑑𝑡𝐺(𝑡)

Here, the pointed brackets in Eq. (6) are the equilibrium statistical mechanical averages involving

the equilibrium distribution function which, in the present case can be written as a product of eqf

the momentum distribution and the three spatial coordinates:

(7) , , , ,eq z zf z p p p p z

can be written as a power series: G t

(8) 1

20 !

n

nn

tG t c

n

Where, is the moment defined as follows:vc thv

(9) vv z zc p iL p

Using a well-known mathematical method known as the Pade' approximant method just the

moments and are sufficient for the calculation.2c 4c

The computation of the moments is straightforward but tedious hence only the final results

will be given:

(10)2 02 124c kT I

l

(11) 4

4 0 2 1 0 3 4 5 64

8 3 2 3c

kTc kT I I I I I Il m

4

The quantities are defined as follows:jI

\*

1 0 2 2

2 22 2

3 0 4 1

2 22 2

5 0 6 1

2 2 2 2cos , cos ,

2 2 2 2cos , cos ,

2 2 2 2sin , sin

z zI K I Kl l l l

z zI K I Kl l l l

z zI K I Kl l l l

MERGEFORMAT (12)

Here, the subscript implies that only the spatial part of the distribution function as defined

in Eq. \* MERGEFORMAT (8) is involved. Before leaving Eq. \* MERGEFORMAT (12) it is

useful to comment on the physical meaning of the quantities designated by the symbol . From jI

Eq. \* MERGEFORMAT (9) the repeated application of the Liouville operator will result in the

force given by the negative gradient of the potential , as shown in Eq. \* ,U z

MERGEFORMAT (4), which will then be followed by higher order derivatives and then thermally

averaged. Ideally, we should calculate all infinite derivatives of the force which is, clearly

impossible and we truncate the expansion \* MERGEFORMAT (8) at thus the seven 4n

quantities averaged in Eq. \* MERGEFORMAT (12) are:

2,2 20,4 2,2 0,2 1,0 1,2

20,1 0,3 1,1

, ; , ; , ; , , , ;

, , ; ,

;a bU z U z U z U z U z U z

U z U z U z

The superscripts are the orders of the differentiation of with respect to and ,U z

respectively.z

The friction coefficient now becomes:

5

(13)𝜁 =

𝜋𝑘𝑇

|𝑐2|3/2

2|𝑐4|

The conductivity, σ and friction coefficient, are related by the standard Nernst-Einstein formula:𝜁

(14)𝜎 = 𝑞 2

𝐿𝑖𝜌𝐿𝑖1𝜁

where is the charge carried by the Li ion, is the number density of the Li ion. Liq Li

2. calculation for Li6.5La2.9Ba0.1Zr1.4Ta0.6O12 at 294 K.∆𝑆

Δ𝑆 = 𝑘ln [𝜋2 2𝑚𝑐|Ψ0|𝑙𝑞 2

𝐿𝑖𝜌𝐿𝑖

𝜎] +𝐸𝑎

𝑇

=

1.3806 𝑥 10 ‒ 23 𝑚2𝑘𝑔𝑆 ‒ 2𝐾 ‒ 2

ln [ (3.14)2 2 𝑥 8.544 𝑥 10 ‒ 11 𝑘𝑔| ‒ 3.21 𝑥 10 ‒ 19𝐽| 8.544 𝑥 10 ‒ 11𝑚 𝑥 (1.602 𝑥 10 ‒ 19 𝐶)2 𝑥 2.49 𝑥 1028 𝑚 ‒ 3

4 𝑥 10 ‒ 2 𝑆𝑚] +

6.25 𝑥 10 ‒ 20𝐽294 𝐾

= 3.63 x 10-22 J/K

6

10 20 30 40 50 60 70 80

A = Ca

A = Sr

Li5La3Ta2O12

A = Ba

Inte

nsity

(a.u

.)

2 (degrees)

Fig. S1. XRD patterns of garnet-type Li6.5La2.9A0.1Zr1.4Ta0.6O12 (A = Ca, Sr, Ba) pellets prepared by SPS process and the calculated pattern of cubic garnet-type Li5La3Ta2O12 (space group: Ia d; 3̅a = 12.80 Å) using the PowderCell program.3

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Fig. S2. SEM images and corresponding elemental mapping of Li6.5La2.9A0.1Zr1.4Ta0.6O12 (A = Ca, Sr, Ba) garnet electrolytes prepared by spark plasma sintering technique.

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Fig. S3. SEM images and corresponding EDS spectra ((grain (1) and grain-boundary (2)) of Li6.5La2.9A0.1Zr1.4Ta0.6O12 (A = Ca, Sr, Ba) garnet electrolytes prepared by spark plasma sintering technique.

References

1 N. Gronbech-jensen, G. Hummer and K. M. Beardmore, Mol. Phys., 1997, 92, 941–946.2 D. A. McQuarrie, Statistical mechanics, Harper & Row, New York, 1976.3 E. J. Cussen, Chem. Commun., 2006, 412–413.