Ionic motion in materials with disordered structures. Module 1 (includes Parts 1&2)

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1 Ionic motion in materials with disordered structures Module 1 (includes Parts 1&2) Klaus Funke, Münster Many-particle dynamics SFB 458 studied from picoseconds to minutes and hours to find rules for ionic motion in disordered materials Glass Tutorial Series: prepared for and produced by the International Material Institute for New Functionality in Glass An NSF sponsored program material herein not for sale Available at Delivered 5/5/06 at Lehigh University

2 Video Module 1 (includes Parts 1 &2) 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

3 An An evolving scheme of of materials science 2. 3a. 3b. 1. 3c. 1. Ideally ordered crystals 2. Point disorder in crystals 3d. 3a. Crystals with structural disorder 3b. Ion-conducting glasses 3c. Polymer electrolytes 3d. Thin-film systems, composite systems

4 level two, random hopping: <Σv i (0). v j (t)> log σ 0 t log ν <Σv i (0). v j (t)> log σ level three, correlated hopping: t log ν

5 dilute strong liquid electrolyte: + + dispersion in σ (ω ) hardly visible solid electrolyte with structural disorder: dispersion over many decades in conductivity

6 Ag 2 S. GeS 2 glassy electrolyte, from 123 K to 473 K 4 log.. 10 (σ T Ω cm/k) SLOPE = 1 SLOPE = K 373 K 323 K 273 K 213 K 183 K 123 K log 10 (ν/hz)

7 Ag 2 S. GeS 2 glassy electrolyte: vibrational contribution removed, set of model spectra included σ ( 0) T and σ ( ) T both Arrhenius activated

8 γ Rb Ag 4 I 5 crystal structure... conductivity spectrum (vibrations removed) at 113 K...

9 Salt-in-polymer electrolyte (1 molal NaPF 6 in a polyurethane) conductivity spectrum (including vibrations) at 303 K... total expt. spectrum, 303 K model spectrum, no vibrations

10 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

11 Measuring conductivity spectra, σ (ω), and permittivity spectra, ε (ω) : Measured quantities: amplitudes and phases of transmitted or reflected waves Basis for evaluation: Maxwell s equations plus boundary conditions σ ( ω ) = iωε ˆ ε ( ω ) σ ( ω) = Re ˆ σ ( ω) and ε( ω) = Re ˆ ε( ω) ˆ 0

12 Ag 2 S. GeS 2 glassy electrolyte, from 123 K to 473 K 4 log.. 10 (σ T Ω cm/k) SLOPE = 1 SLOPE = K 373 K 323 K 273 K 213 K 183 K 123 K log 10 (ν/hz)

13 Conductivity isotherms for glassy 0.3 Li 2 O. 0.7 B 2 O 3 λ = 1

14 Gradual transition into NCL behavior : NCL Scaled conductivity spectrum, for various crystalline and glassy electrolytes NCL: Nearly Constant Loss λ = 1 slope = 1

15 0.2 Na 2 O. 0.8 GeO 2 glass, permittivity isotherms λ = 1

16 Scaled permittivity for 0.2 Na 2 O. 0.8 GeO 2 glass ε S ( ω ) S = ω ε ( ε ( ω S ) ε ( ) ) σ ( 0) 1 ω 0 0 = S Im σˆ S ( ω ) S λ = 1

17 Real and imaginary parts of dielectric modulus do NOT scale 1 M ˆ ( ω ) = M '( ω ) + im ''( ω ) = ˆ ε ( ) ω M (ω) M (ω)

18 This lecture continues on a 2 nd module - Video Module 2 : Dynamics of Ions (Part 3)

19 Ionic motion in materials with disordered structures Video Module 2 (Part 3- Dynamics of Ions) Klaus Funke, Münster Many-particle dynamics SFB 458 studied from picoseconds to minutes and hours to find rules for ionic motion in disordered materials Glass Tutorial Series: prepared for and produced by the International Material Institute for New Functionality in Glass An NSF sponsored program material herein not for sale Available at Delivered 5/5/06 at Lehigh University

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

21 Introducing the time-dependent correlation factor, W(t) 0 σ log σ ( ω ) ( ) <v(0). v(t)> t log ω 0 log W () t σ ( ω ) σ = + 0 ( ) () t cos( ω t) 1 W& dt log t ˆ σ σ ( ω) ( ) = 1+ HOP 0 W& t () exp( iω t) dt

22 logσ S ( ) ω ( t) = log( W( t) W( ) ) logw S / 0 log ω 0 log t ˆ σ S ( ω) ˆ σ = σ HOP ( ω) ( 0) = 1+ iω ( W () t 1 ) exp ( i t ) S ω 0 dt As soon as W S (t) is known, the scaled conductivity is also known. Very realistic results are obtained by the MIGRATION concept, the acronym standing for MIsmatch Generated Relaxation for the Accommodation and Transport of IONs In the model, ω 0 marks the onset of the dispersion. Scaled time: t S = tω 0, scaled frequency: ω S = ω /ω 0

23 single-particle route 1 W(t) = time-dependent correlation factor many-particle route 0 t W ( ) W& W g& g () t () t () t () t = B g& = Γ 0 W () t [ + Γ 0 ] if motion is completely localised () t N () t W(t) = correlation factor W(t) and W S (t S ) g(t) = mismatch function σ(ω) and σ S (ω S ) N () t N ( ) = ( B g () t ) λ N(t) = number function ε(ω) and ε S (ω S )

24 In 0.4 Ca(NO 3 ) KNO 3 (CKN) above T G T T K = λ + 1 changes from 2.0 to 1.2 T G

25 Ag 2 S. GeS 2 glassy electrolyte: vibrational contribution removed, set of model spectra included σ ( 0) T and σ ( ) T both Arrhenius activated

26 Conductivity isotherms for glassy 0.3 Li 2 O. 0.7 B 2 O 3 λ = 1

27 Gradual transition into NCL behavior : NCL Scaled conductivity spectrum, for various crystalline and glassy electrolytes NCL: Nearly Constant Loss λ = 1 slope = 1

28 Scaled permittivity for 0.2 Na 2 O. 0.8 GeO 2 glass ε S ( ω ) S = ω ε ( ε ( ω S ) ε ( ) ) σ ( 0) 1 ω 0 0 = S Im σˆ S ( ω ) S λ = 1

29 Localized mean square displacement for glasses x Na 2 O (1-x) GeO 2 Therefore: r 2 LOC ε S 2 () t = r () t 6 Dt = 6 D ( W () t ' 1) ω 6D d dt 0 2 ( ω ) FT r () t LOC t 0 and S ε S dt ' ω 6 D 0 2 ( ) = r ( ) 0 LOC 0 log r () t 6D 2 ω x = 0.2 log ω 0 6 D r 2 LOC () t ( t ) log ω 0 r 2 LOC ( ) 0.5 1/ 3 x and about 65 % of average Na - Na distance

30 Treatment of localised motion: W& W NCL NCL () t () t = B NCL g& () t + Γ 0 rate: Γ 0 g& g () t () t =... as before, with g(t) close to 1 log σ 0 W NCL () t = Γ ( ω end + Γ0 ) exp ( Γ0t ) ω end 0 1 ω end = Γ B 2 0 NCL 2 ω end i.e., Debye for B NCL = 0 Γ 0 log ω

31 Conductivity spectrum of RbAg 4 I 5, below the K first order β γ phase transition, after removal of the vibrational component NCL NCL : Nearly Constant Loss ν 1 = Γ 0 /2π ν 2 = ω end /2π

32 M. Jansen R. Dinnebier A. Fitch Structure of γ Rb Ag 4 I 5

33 Conductivity spectrum of a silver iodide - silver metaphosphate glass, after removal of frequency-squared vibrational component NCL We now find : Γ 0 / 2π << (Γ 0 / 2π) loc = ν 1, ν 1 being activated with only 0.05 ev. Localised motion is ubiquitous, e.g., Ag + - nbo - dipoles

34 Conductivity spectrum of NaPF 6 (1 molal) in a polyurethane, prepared by crosslinking a tri-functional random PEO : PPO copolymer, 4 : 1 by mol, M W ~ 3600 g / mol T = 303 K VIB NCL MIG NCL

35 Conclusion log σ log σ log σ 1 2 MIG NCL VIB < 1 log ω log ω log ω Ionic materials with disordered structures (crystalline, glassy, polymeric) always show this kind of superposition caused by potentially translational hopping motion strictly localised non-vib. motion vibrational motion all three of them being collective and cooperative in character.

36 Thanks to : Deutsche Forschungsgemeinschaft for SFB 458 M.D. Ingram, Aberdeen M. Jansen, Stuttgart R. Dinnebier, Stuttgart A. Fitch, Grenoble A.S. Nowick, Columbia H. Jain, Lehigh A. Heuer, Münster R.D. Banhatti, Münster C. Cramer, Münster D. Wilmer, Münster P. Singh, Münster S.J. Pas, Münster R. Belin, Montpellier / Münster THANK YOU FOR YOUR ATTENTION

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