Takuji Oda, Yasuhisa Oya, Satoru Tanaka

The University of Tokyo

Why Li2O?

Li2SiO3 Li4SiO4 (*Li: 50%) Li2TiO3 LiAlO2

Li2O

Important features of Li-containing oxides are:(i) Lightness of Li(ii) High mobility of Li

, which are seen also in Li2O.

The simple structure facilitates understanding of phenomena, and has advantages in computer simulations.

Background: tritium in Li-containing oxidesFusion reactor

PlasmaMaterials

Blanket

Tritium““Tritium in LiTritium in Li22OO””Surface processesExistence statesDiffusion behaviorRadiation response

Defect formation/annihilationDefect clustering

Interaction with defectsLi vacancyF centers

Design

Stability

Quantum mechanical calc.

Dynamics for ~ ns (for low diff. Energy) Molecular dynamics sim. (MD)

Collision in the dynamics for ~ps

MD

Interaction/trapping forceQuantum mechanical calc.

““AtomicAtomic--scale understandingsscale understandings””

[The present purpose] Model diffusion of Li+ in defective Li2O.

Model tritium behavior in reactor conditions, based on atomic-scale understandings.

Approach: multi-scale modeling

1 2

3

(1) Evaluate diffusion behavior in defective Li2O under dynamics, using MD.(2) Compare results with a classical theory, to evaluate effects of the dynamics.(3) Refine model parameters, using quantum mechanical calculation.

Calculation method-1/3: Molecular Dynamics (MD)① Three variables are softly fixed to realize a certain thermo-　　　 　

dynamical state, namely ensemble.e.g. NPT, NEV

②　Coordinates/force/velocity on each atom are updated step by stepbased on equation of motion under predefined potential model.

③ 　Thermo-chemical/physical property is derived from variations of coordinates/force/velocity, based on the statistics.

)()()(2)( 42 ttm

ttttt ∆Ο+∆+∆−−=∆+ Frrr

①

② ③ MD can treat 106 atoms for ~ µs.

Applicable for study on diffusion.

Relatively low calculation cost.

If the diffusion consists of a mono process, it can be written by a classical theory:

pfdDCT ××= 6

2 d: a movement by a jump (m) f: correlation factorp: number of jumps (s-1)

Then, by converting the equation into a form with activation energies,

−×××××=RTEFreq

NN

nfdD datoms

defectpathCT exp.][6

2 npath: number of migration path [Freq.] : attempt frequency (s-1)Ed: diffusion barrier (eV)

Effects of the dynamics, which are not included in this equation,can be estimated by comparison between DMD and DCT.

In MD, diffusion coefficients can be determined by Einstein’s relation:

{ }

atom

N

iMD N

rrD

atom

∑=

−×=×= 1

2

2)0()(

61

61

τ

τλ

τ

λ: mean free path (m)τ: mean free time (s)Natom: number of atoms in the system

Calculation method-2/3: description of diffusionMD

Classical theory

Calculation method-3/3: quantum calculation

2x2x2

Li : O :

Conventional cell (Li8O4) 2x2x2 supercell (Li64O32)

Calculation was performed based ondensity functional theory (DFT) with GGA-PBE functional and plane-wave basis set, using CASTEP code.

K-point grid: 3x3x3Energy cutoff: 380 eV

Quantum calculation gives us information of electronic statesvia wave function.

Charged defects.Optical property.Precise energy calculation.

Outline of results

① [1] Li diffusivities in defective Li2O, by MD.

② [2~8] Parameters in the classical theory, by MD/MS.

③ [9] Temperature dependence of some parameters, by MD/MS.

④ [10] “Effects of the dynamics”, by comparison of DMD & DCT.

⑤ [11,12] How to refine model parameters, by quantum calculation.

Result-1： Li diffusivity in Li2O by MD

0.0003 0.0006 0.0009 0.0012 0.0015 0.0018-32

-28

-24

-20

-16

1200 K

2400 K

(a)

(b)

(c)

log

( D

Li )

/

(m2 s

-1)

Reciprocal temperature / K-1

Li1000O500 Li999O500

(d)

1600 K

Fig.1: Li diffusivities in Li2O (Li1000O500 and Li999O500)(MD by DL_POLY code under NpT ensemble)

(a) Extrinsic region⇒ assisted by defects

① 0.1% Li vac. (Li999O500)② 0.1% Li int. (Li1001O500)

(b) Intrinsic region-1(c) Intrinsic region-2

* the boundary is SI critical temp.(d) Liquid state

Meanings of each parameter

⑤ Ed : diffusion barrier

0.00

0.05

0.10

0.15

0.20

0.25

⊿ E

/ e

V

Path coordinates

① d : unit jump distance ② npath : number of migration path ③ f : correlation factor

④ [Freq.] : attempt frequency

−×

××××=RTEFreq

NN

fndD datoms

defectpathCT exp.][6

2

Result-2：　d (unit migration distance)

Fig.3: Temperature dependence 　　 of lattice parameter (a)

Fig.1: Li vac. movement (0.5a)

Li vac.

0 500 1000 1500 2000 25004.5

4.6

4.7

4.8

4.9

5.0

5.1

Latti

ce c

onst

ant (

a) /

Å

Temperature / K

MD results

dvac. ~ 0.23 nmdint. ~ 0.20 nm

(error < 5 %)Fig.2: Li inｔ. movement (0.433a)

Li

O

Li int.

Result-3：　npath (migration path) & f (correlation factor)

Fig.2: Possible jumps of Li int. (npath = 8)

0 200000 4000000.0

0.5

1.0

1.5

2.0

2.5

Vacancy

Interstitial

Cor

rela

tion

fact

or

Number of jumpsFig.3: Determination by Monte Carlo simulation

f = 0.66* for vacancyf = 2.0 for interstitial

*0.653 in analytical(error ~ 1%)

Fig.1: Possible jumps of Li vac. (npath= 6)

Li vac.Li int.

Result-4：　[Freq.] (attempt frequency)

[Freq.] can be determined from a changing point of Li motion.

[Freq.] = 0.5 x [changing point (s-1)]

Fig.2: Estimation of [Freq.] by MD

[Freq.]=16 ps-1 (±1) for Li [Freq.]=15 ps-1 (±1) for O

(error ~ 5%)

Fig.1: Typical Li+ movement

0 10 20 30 4020

25

30

35

40

2*[fr

eq.]

/ p

s-1

Simulation time / ps

Li_x, Li_y, Li_z O_x, O_y, O_z

*These values were obtained in dynamics calculation.

−×

××××=RTEFreq

NN

fndD datoms

defectpathCT exp.][6

2

Result-5：　Ed for Li vac. (diffusion barrier)

0.00

0.05

0.10

0.15

0.20

0.25

③②①

⊿ E

/ e

V

Fig.1: ①Initial (left), ②transition (middle) & ③final (right) states for Li vac. migration(Li: , O: , migrating Li: )

Fig.2: Migration energy (Ed) by MS

Tab. 1 Ed for Li vacancy

a T. Matsuo et al., J. Chem. Soc. 79 (1983) 1205.b H. Ohno et al, J. Nucl. Mater. 118 (1983) 242.

① ② ③

Experiment 0.38 eV a , 0.42 eVb

QM 0.237 eVMS 0.24 eV

Result-6：　 Ed for Li int. (diffusion barrier)

Tab. 1: Ed for Li interstitial

0 eV

1 eV MD/MS

Fig.2: Migration path

QM 0.529 eVMS 0.59 eV

Fig.1: ①Initial (left), ②transition (middle) & ③final (right) states for Li int. migration(Li: , O: , migrating Li: )

① ② ③

①

②③

Result-7：　summary of parameters

Tab.1: Summary of parameters determining Li diffusivity

① Li vac. (Li999O500), ② Li int. (Li1001O500)

npath : number of migration path f : correlation factor d : unit jump distanceEd : diffusion barrier[Freq.] : attempt frequency

① Li vac. ② Li int.Li999O500 Li1001O500

d / nm 0.23 0.2 5npath 6 8 -

f 0.65 2 -[Ndefect/Natom] 0.001 0.001 -

[Freq.] / ps-1 16 15 5Ed / eV 0.23 0.56 5

(error / %)

−×

××××=RTEFreq

NN

fndD datoms

defectpathCT exp.][6

2

Result-8： comparison between DMD and DCT

Fig.1: Comparison between MD results and the classical theory estimation

① Li vac. : DCT > DMD, ② Li int. : DCT < DMD

How about temperature dependences of [Freq.] and Ed ?

0.0006 0.0008 0.0010 0.0012 0.0014 0.0016-32

-28

-24

-20

-16

Ln (

DLi )

/ (

m2 s

-1)

Temperature / K-1

DMD of Li vac. DCT of Li vac. DMD of Li int. DCT of Li int.

−×

××××=RTEFreq

NNdfnD datoms

defectpathLi exp.][6

2

Differences between DMD and DCT.

Result-9：　temperature dependence of [Freq.] and Ed

1.00 1.01 1.02 1.03 1.04 1.050.00

0.05

0.10

0.15

0.20

0.25

Li vac.

Diff

usio

n ba

rrie

r (E d

) /

eV

Linear expansion ratio (a/a0)

Results by MS Ed(x) = -10.13 * x

2 + 17.83 * x -7.48

Fig.1: Temp. dependence of [Freq.]

0 300 600 900 1200 150015.4

15.6

15.8

16.0

[Fre

q.]

/

ps-1

Temperature / K

Li [100] Li [010] Li [001]

[Freq.]: weak dependenceEd: strong dependence

Since Ed is an index of exponential function and thus influential for Dct, its temperature dependence should be considered.

−×××××=RTEFreq

NN

nfdD datoms

defectpathCT exp.][6

2

Fig.2: Temp. dependence of Ed

Result-10：　re-comparison between DMD and revised DCT

0.0006 0.0008 0.0010 0.0012 0.0014 0.0016-32

-28

-24

-20

-16

Ln ( D

Li )

/ (m

2 s-

1 )

Reciprocal temperature / K-1

DMD-vac., DCT1-vac., DCT2-vac. D

MD-int., D

CT1-int., D

CT2-int.

Fig.1: Comparison between MD (DMD) and the classical theory (DCT)（CT2 considers temperature dependence of Ed, while CT1 do not.）

By taking into account the temperature dependence of Ed,, the classical theory 　overestimates diffusion coefficients. (= “effects of the dynamics”)

The overestimation is about 0～1 order of magnitude in diffusion coefficients.

Result-11：　refinement of Ed by quantum calculation

Fig. 1: Comparison in temperature dependences of Ed for Li vacancy

1.000 1.005 1.010 1.015 1.020 1.025 1.0300.00

0.05

0.10

0.15

0.20

0.25

0.30

Diff

usio

n ba

rrier

(Ed)

/ e

V

Linear expansion coef.

Quantum FIT-HF FIT-LDA FIT-GGA FIT-EMP

Temperature dependence of Ed is similar between quantum calculation and MS.

Based on the quantum results, Ed can be refined.

Table 1: Comparison of [Freq.]

-4.0x10-11 0.0 4.0x10-110.0

4.0x10-20

8.0x10-20

1.2x10-19

∆ Energ

y /

J

Displacement / m

FIT-LDA(MS)

fitting: 0.5×119.68 x2 16.26 ps-1 for Li(6.9)

-4.0x10-11 0.0 4.0x10-110.0

4.0x10-20

8.0x10-20

1.2x10-19

∆ Energ

y /

J

Displacement / m

DFT

fitting: 0.5×92.39 x2 14.3 ps-1 for Li(6.9)

Fig.1: [Freq.] in the potential model Fig.2: [Freq.] in quantum calc.

MD 15.8 ps-1

MS 16.3 ps-1

DFT MS 14.3 ps-1

FIT-LDA

Since [Freq.] was obtained in the dynamics, an alternative way to evaluate itin the statics is needed for refinement. >> harmonic oscillator

Result-12：　refinement of [Freq.] by quantum calculation

[Freq.] seems to be overestimated by about 10 %.(This may be negligible)

Summary

Temperature dependence is negligible.

Temperature dependence is essential.

Diffusion coefficients are overestimated 　　　　　 by 0~1 order of magnitude.

Li diffusion behaviors in defective Li2O were modeled by comparison of MD results and estimations based on a classical theory.

−×

×××××=RTTEFreq

NN

fndD datoms

defectpathCT

)(exp.][6

2

α

α (effects of the dynamics) is ~10.d, npth and f can be absolutely determined from the crystalline structure.[Freq.] and Ed can be refined using quantum mechanics calculation.