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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
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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)
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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
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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
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dilute strongliquid electrolyte:
solid electrolytewith structuraldisorder:
+dispersionin σ (ω )hardlyvisible
dispersionover manydecades inconductivity
− +
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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
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Ag2S . GeS2 glassy electrolyte:vibrational contribution removed, set of model spectra included
( ) T⋅0σ ( ) T⋅∞σ both Arrhenius activatedand
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γ − Rb Ag4 I5
crystal structure .......
conductivity spectrum(vibrations removed)at 113 K ....................
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conductivity spectrum(including vibrations)at 303 K .....................
Salt-in-polymerelectrolyte(1 molal NaPF6 ina polyurethane)
total expt. spectrum, 303 K
model spectrum,no vibrations
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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
![Page 11: Ionic motion in materials with disordered structures ...inimif/teched/LecBasic/Tut_FunkeAB.pdfAn evolving scheme of materials science 2. 3a. 3b. 3c. 3d. 1. 1. Ideally ordered crystals](https://reader033.vdocuments.us/reader033/viewer/2022060521/60501ae87792343428645bac/html5/thumbnails/11.jpg)
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=
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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
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Conductivity isotherms for glassy 0.3 Li2O . 0.7 B2O3
λ = 1
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Scaled conductivityspectrum, for variouscrystalline and glassyelectrolytes
λ = 1
slope = 1
Gradual transition into NCL behavior :
NCL
NCL:Nearly Constant Loss
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0.2 Na2O . 0.8 GeO2 glass, permittivity isotherms
λ = 1
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( ) ( ) ( )( )( ) ( )SS
S
SSS ωσ
ωσεωεεωωε ˆIm1
000 ⋅=
∞−⋅⋅=
Scaled permittivity for 0.2 Na2O . 0.8 GeO2 glass
λ = 1
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Real and imaginary parts of dielectric modulus do NOT scale
( ) ( ) ( ) ( )ωεωωω
ˆ1'''ˆ =+= iMMM
M″ (ω)
M′ (ω)
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This lecture continues on a 2nd module -
Video Module 2 : Dynamics of Ions (Part 3)
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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
![Page 20: Ionic motion in materials with disordered structures ...inimif/teched/LecBasic/Tut_FunkeAB.pdfAn evolving scheme of materials science 2. 3a. 3b. 3c. 3d. 1. 1. Ideally ordered crystals](https://reader033.vdocuments.us/reader033/viewer/2022060521/60501ae87792343428645bac/html5/thumbnails/20.jpg)
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)
![Page 21: Ionic motion in materials with disordered structures ...inimif/teched/LecBasic/Tut_FunkeAB.pdfAn evolving scheme of materials science 2. 3a. 3b. 3c. 3d. 1. 1. Ideally ordered crystals](https://reader033.vdocuments.us/reader033/viewer/2022060521/60501ae87792343428645bac/html5/thumbnails/21.jpg)
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)>
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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
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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
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TG
TT
In 0.4 Ca(NO3)2 0.6 KNO3 (CKN)
above TG
K = λ + 1 changes
from 2.0 to 1.2
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Ag2S . GeS2 glassy electrolyte:vibrational contribution removed, set of model spectra included
( ) T⋅0σ ( ) T⋅∞σ both Arrhenius activatedand
![Page 26: Ionic motion in materials with disordered structures ...inimif/teched/LecBasic/Tut_FunkeAB.pdfAn evolving scheme of materials science 2. 3a. 3b. 3c. 3d. 1. 1. Ideally ordered crystals](https://reader033.vdocuments.us/reader033/viewer/2022060521/60501ae87792343428645bac/html5/thumbnails/26.jpg)
Conductivity isotherms for glassy 0.3 Li2O . 0.7 B2O3
λ = 1
![Page 27: Ionic motion in materials with disordered structures ...inimif/teched/LecBasic/Tut_FunkeAB.pdfAn evolving scheme of materials science 2. 3a. 3b. 3c. 3d. 1. 1. Ideally ordered crystals](https://reader033.vdocuments.us/reader033/viewer/2022060521/60501ae87792343428645bac/html5/thumbnails/27.jpg)
Scaled conductivityspectrum, for variouscrystalline and glassyelectrolytes
λ = 1
slope = 1
Gradual transition into NCL behavior :
NCL
NCL:Nearly Constant Loss
![Page 28: Ionic motion in materials with disordered structures ...inimif/teched/LecBasic/Tut_FunkeAB.pdfAn evolving scheme of materials science 2. 3a. 3b. 3c. 3d. 1. 1. Ideally ordered crystals](https://reader033.vdocuments.us/reader033/viewer/2022060521/60501ae87792343428645bac/html5/thumbnails/28.jpg)
( ) ( ) ( )( )( ) ( )SS
S
SSS ωσ
ωσεωεεωωε ˆIm1
000 ⋅=
∞−⋅⋅=
Scaled permittivity for 0.2 Na2O . 0.8 GeO2 glass
λ = 1
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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:
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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
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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
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Structure of γ − Rb Ag4 I5
M. JansenR. DinnebierA. Fitch
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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
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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
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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
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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
THANK YOU FOR YOUR ATTENTION