Klaus Funke, Münster

Ionic motionin materials with disordered structures

Module 1 (includes Parts 1&2)

Many-particle dynamics

studied from picoseconds

to minutes and hours to

find rules for ionic motion

in disordered materials

SFB 458

Glass Tutorial Series: prepared for and produced by theInternational Material Institute for New Functionality in GlassAn NSF sponsored program – material herein not for sale Available at www.lehigh.edu/imi

Delivered 5/5/06 at Lehigh University

1. Materials and phenomena : the rôle of disorder

2. Microscopy in time : σ(ω), ε(ω) and scaling properties

3. Dynamics of ions in motion : searching for simple rules

Video Module 1 (includes Parts 1 &2)

An evolving scheme of materials science

2.

3a.

3b.

3c.

3d.

1.

1. Ideally ordered crystals2. Point disorder in crystals

3a. Crystals with structural disorder

3b. Ion-conducting glasses

3c. Polymer electrolytes3d. Thin-film systems,

composite systems

An evolving scheme of materials science

level two,random hopping:

level three,correlated hopping:

0 t

<Σvi(0).vj(t)>

log ν

log σ

t

<Σvi(0).vj(t)> log σ

log ν

dilute strongliquid electrolyte:

solid electrolytewith structuraldisorder:

+dispersionin σ (ω )hardlyvisible

dispersionover manydecades inconductivity

− +

0 2 4 6 8 10 12 14-8

-6

-4

-2

0

2

4

SLOPE = 1

SLOPE = 2

473 K 373 K 323 K 273 K 213 K 183 K 123 K

..

log 10

(σ T

Ω c

m/K

)

log10(ν/Hz)

Ag2S . GeS2 glassy electrolyte, from 123 K to 473 K

Ag2S . GeS2 glassy electrolyte:vibrational contribution removed, set of model spectra included

( ) T⋅0σ ( ) T⋅∞σ both Arrhenius activatedand

γ − Rb Ag4 I5

crystal structure .......

conductivity spectrum(vibrations removed)at 113 K ....................

conductivity spectrum(including vibrations)at 303 K .....................

Salt-in-polymerelectrolyte(1 molal NaPF6 ina polyurethane)

total expt. spectrum, 303 K

model spectrum,no vibrations

1. Materials and phenomena : the rôle of disorder

2. Microscopy in time : σ(ω), ε(ω) and scaling properties

3. Dynamics of ions in motion : searching for simple rules

Measuring conductivity spectra, σ (ω), and permittivity spectra, ε (ω) :

Measured quantities: amplitudes and phases of transmitted or reflected wavesBasis for evaluation: Maxwell‘s equations plus boundary conditions

( ) ( )ωεωεωσ ˆˆ 0i= ( ) ( )ωσωσ ˆRe= and ( ) ( )ωεωε ˆRe=

0 2 4 6 8 10 12 14-8

-6

-4

-2

0

2

4

SLOPE = 1

SLOPE = 2

473 K 373 K 323 K 273 K 213 K 183 K 123 K

..

log 10

(σ T

Ω c

m/K

)

log10(ν/Hz)

Ag2S . GeS2 glassy electrolyte, from 123 K to 473 K

Conductivity isotherms for glassy 0.3 Li2O . 0.7 B2O3

λ = 1

Scaled conductivityspectrum, for variouscrystalline and glassyelectrolytes

λ = 1

slope = 1

Gradual transition into NCL behavior :

NCL

NCL:Nearly Constant Loss

0.2 Na2O . 0.8 GeO2 glass, permittivity isotherms

λ = 1

( ) ( ) ( )( )( ) ( )SS

S

SSS ωσ

ωσεωεεωωε ˆIm1

000 ⋅=

∞−⋅⋅=

Scaled permittivity for 0.2 Na2O . 0.8 GeO2 glass

λ = 1

Real and imaginary parts of dielectric modulus do NOT scale

( ) ( ) ( ) ( )ωεωωω

ˆ1'''ˆ =+= iMMM

M″ (ω)

M′ (ω)

This lecture continues on a 2nd module -

Video Module 2 : Dynamics of Ions (Part 3)

Klaus Funke, Münster

Ionic motionin materials with disordered structures

Video Module 2 (Part 3- Dynamics of Ions)

Many-particle dynamics

studied from picoseconds

to minutes and hours to

find rules for ionic motion

in disordered materials

SFB 458

Glass Tutorial Series: prepared for and produced by theInternational Material Institute for New Functionality in GlassAn NSF sponsored program – material herein not for sale Available at www.lehigh.edu/imi

Delivered 5/5/06 at Lehigh University

1. Materials and phenomena : the rôle of disorder

2. Microscopy in time : σ(ω), ε(ω) and scaling properties

3. Dynamics of ions in motion : searching for simple rules

Video Module 2 : Dynamics of Ions (includes Part 3)

Introducing the time-dependent correlation factor, W(t)

t

log t

log ω

( )( )∞σωσlog

( )tWlog ( )( ) ( ) ( )∫

∞

⋅⋅⋅+=∞ 0

cos1 dtttW ωσ

ωσ &

( )( ) ( ) ( )∫

∞

⋅⋅−⋅+=∞ 0

exp1ˆ

dttitWHOP ωσ

ωσ &

0

0

<v(0).v(t)>

ωlog

0

( )ωσ Slog

0

( ) ( ) ( )( )∞= WtWtWS /loglog

tlog

( ) ( )( ) ( )( ) ( )∫

∞

⋅⋅−⋅−+==0

exp110

ˆˆ dttitWi SHOP

S ωωσ

ωσωσ

As soon as WS(t) is known, the scaled conductivity is also known. Very realis-tic results are obtained by the MIGRATION concept, the acronym standing for MIsmatch Generated Relaxation for the Accommodation and Transport of IONs

In the model, ω0 marks the onset of the dispersion. Scaled time: tS = tω0 , scaled frequency: ωS = ω /ω0

single-particle route

many-particle route t

W(t) time-dependent correlation factor1

0

=

( )∞W

( )( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )( )λtgBNtN

tNtWtgtg

tgBtWtW

⋅=∞−

⋅⋅Γ=−

⋅−=−

0&

&&

W(t) = correlation factor

N(t) = number function

g(t) = mismatch function

W(t) and WS(tS)

σ(ω) and σS(ωS)

ε(ω) and εS(ωS)

[ + Γ0] if motion is completely localised

TG

TT

In 0.4 Ca(NO3)2 0.6 KNO3 (CKN)

above TG

K = λ + 1 changes

from 2.0 to 1.2

Ag2S . GeS2 glassy electrolyte:vibrational contribution removed, set of model spectra included

( ) T⋅0σ ( ) T⋅∞σ both Arrhenius activatedand

Conductivity isotherms for glassy 0.3 Li2O . 0.7 B2O3

λ = 1

Scaled conductivityspectrum, for variouscrystalline and glassyelectrolytes

λ = 1

slope = 1

Gradual transition into NCL behavior :

NCL

NCL:Nearly Constant Loss

( ) ( ) ( )( )( ) ( )SS

S

SSS ωσ

ωσεωεεωωε ˆIm1

000 ⋅=

∞−⋅⋅=

Scaled permittivity for 0.2 Na2O . 0.8 GeO2 glass

λ = 1

Localized mean square displacement for glasses x Na2O (1-x) GeO2

( ) ⎟⎠⎞

⎜⎝⎛ ⋅

Dtr

6log 02 ω

( ) ⎟⎠⎞

⎜⎝⎛ ⋅ tr

D LOC20

6log ω

( ) ( ) ( )( )∫ −⋅=−=t

SLOC dttWDDttrtr0

22 '1'66

( ) ( )∞= 20

60 LOCS r

Dωε

( ) 5.02 ∞LOCr about 65 % of average Na - Na distance3/1−∝ x and

x = 0.2

( )0log ω⋅t

( ) ( ) ⎥⎦⎤

⎢⎣⎡⋅≈ tr

dtdFT

D LOCS20

6ωωε andTherefore:

log ω

log σ

2

1

0

0Γ

endω

Treatment of localised motion:

( )( ) ( )

( )( ) .....

0

=−

Γ+⋅−=−

tgtg

tgBtWtW

NCLNCL

NCL

&

&&

( ) ( ) ( ) endendNCL t

tWωω −ΓΓ+

Γ=

00

0

exp

20 NCLend B⋅Γ=ω

i.e., Debye for 0=NCLB

as before, with g(t) close to 1

rate: Γ0

Conductivity spectrum of RbAg4I5, below the 121.8 K first order β−γ phase transition, after removal of the vibrational component

ν1 = Γ0/2π ν2 = ωend/2π

NCL

NCL :Nearly Constant Loss

Structure of γ − Rb Ag4 I5

M. JansenR. DinnebierA. Fitch

Conductivity spectrum of a silver iodide - silver metaphosphate glass,after removal of frequency-squared vibrational component

We now find : Γ0 / 2π << (Γ0 / 2π)loc= ν1 , ν1 being activated with only 0.05 eV.

NCL

Localised motion is ubiquitous, e.g., Ag+ - nBO- dipoles

Conductivity spectrum of NaPF6 (1 molal) in a polyurethane,

prepared by crosslinking a tri-functional random PEO : PPO copolymer,

4 : 1 by mol, MW ~ 3600 g / mol

T = 303 K

MIG

NCL

VIB

NCL

MIG NCL VIB

log σ log σ log σ

Conclusion

log ω log ω log ω

Ionic materials with disordered structures(crystalline, glassy, polymeric)

always show this kind of superpositioncaused by

potentially translationalhopping motion

strictly localisednon-vib. motion

vibrationalmotion

all three of them being collective and cooperative in character.

21

< 1

Thanks to : Deutsche Forschungsgemeinschaft for SFB 458

M.D. Ingram, AberdeenM. Jansen, StuttgartR. Dinnebier, StuttgartA. Fitch, GrenobleA.S. Nowick, ColumbiaH. Jain, LehighA. Heuer, Münster

R.D. Banhatti, MünsterC. Cramer, MünsterD. Wilmer, MünsterP. Singh, MünsterS.J. Pas, MünsterR. Belin, Montpellier / Münster

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