L2: Solid Electrolytes

Professor S.R. Elliott and Dr S.N. Taraskin

Lecture 1: Structure of Disordered Materials

Lecture Synopsis• Lecture 1: Structure of Disordered Materials (SRE)

• Lecture 2: Defects (SRE)

• Lecture 3: Examples of Solid Electrolytes (SRE)

• Lecture 4: Mechanisms of Diffusion (SNT)

• Lecture 5: Fundamentals of Diffusion (SNT)

• Lecture 6,7: Random Walk and Approximations (SNT)

• Lecture 8: Conductivity and Disorder (SNT)

• Lecture 9: Depolarization Current (SNT)

• Lecture 10: Impedance Spectroscopy and Radiotracers (SRE)

• Lecture 11: NMR and QENS (SRE)

• Lecture 12: Applications (SRE)

• Solid electrolytes - or superionic conductors - or fast-ion conductors - are materials in which the ionic (cationic/anionic) conductivity is comparable to that of liquid electrolytes, and the electronic conductivity is negligible

• In solids, atomic diffusion can only occur via structural defects - so need to understand disorder

• Disorder is defined with respect to a reference structure - eg the ideal crystal

Salvador Dali (1952)

“Galateaof theSpheres”

The Physics & Chemistry of Solids

SRE

Crystals

Ruavb wc

•An ideal crystal is generated by the translationally periodic repeat of a unit cell.•A lattice is an infinite array of mathematical points having the translational periodicity of the crystal•The unit cell is defined by the vectors a, b, c (in 3D)•Any 2 lattice points are connected by the vector

•Unit cells can be: primitive (P), body-centred (I), side-centred (C), face-centred (F)

Disordered Materials• Two classes of materials have topologically disordered structures:

liquids & glasses

Liquids• Topological disorder plus dynamical disorder (orientational and

translational)

Topological Spin

Substitutional Positional/vibrational

Types of disorder

• Slow cooling of a melt -> crystallization- crystal nuclei grow into stable crystallites in liquid, before the viscosity (exponentially) increases with decreasing temperature

• Crystallization of melt is a first-order thermodynamic phase transition (density discontinuity)

Crystallization

Glasses/Amorphous Solids• Topological (quenched) disorder

• Rapid cooling of a melt supercooled liquid- rapid increase with viscosity with decreasing temperature

precludes crystal nucleation and growth transforms to a glass at Tg (glass-transition temperature)

glass = a “solid” liquid on the experimental timescale amorphous = non-crystalline

Vitrification

Structure of Liquids and Glasses• Structure has no long-range order (LRO): no lattice, no unit cell.• But structure is (generally) orientationally isotropic • Statistical description in terms of spatial correlation functions- e.g. pair distribution functions, i.e. the atomic density function, ρ(r)

- Radial distribution function (RDF) is defined as:

J r 4r 2 r -it is the average probability of finding an atom in the distance interval r → r + dr from a given atom at r = 0

• Area under an RDF peak gives the average atomic coordination number• Positions of RDF peaks give radii of neighbour shells

Short-range Order (SRO)• Absence of LRO in glasses and liquids does not mean that there

is no structural order whatsoever• Glasses and liquids can have (varying degrees of) SRO

- e.g. corner (O)-shared SiO4 tetrahedra in g-SiO2

- well-defined coordination numbers, bond lengths, bond angles

• Bond-length and bond-angle disorder (c.f. widths of first 2 RDF peaks) are not generally sufficient alone to destroy LRO

• Instead, fluctuations in the dihedral angle, φ, destroy correlations between nearest-neighbour structural units for covalent systems.

P( )

ideal amorphous

ideal crystal

Dihedral angle

Structure Determination of Glasses and Liquids

n 2dhkl sin

•X-ray, neutron or electron diffraction (scattering plus interference) is commonly used to study the atomic structure of materials•For the case of crystals, with well-defined lattice planes (hkl), Bragg diffraction occurs:

• A crystal acts like a 3D diffraction grating for X-rays etc, and diffracted beams only lie in particular directions, at angles θhkl

• For amorphous materials, diffraction occurs over a range of angles- Bragg spots (for crystals) become diffuse haloes (for glasses)

hkl

• Complex amplitude of a wave, scattered by atom i, is:

Fi = fi exp [i(k − k0). ri] ,

where

K k k0 4 sin

• The atomic scattering factor, fi, is constant for neutrons - strongly decreasing function of K or θ for X-rays & e-

wavevector transfer

Nb k0 is incident wavevector k is scattered wavevector

• Total scattering intensity from a collection of (pairs of) atoms (i,j) is generally:

i fi fj exp i k k0 .rij

j

where

rij ri rj

• Amorphous materials are spatially isotropic overall. The orientational average of the phase factor is:

exp i k k0 .rij 1

4rij2 exp iKrij cos

0

2rij2 sind

sin Krij

Krij

,

giving the Debye scattering equation:

I i fi fj

sin KrijKrijj

• For a monatomic solid (fi = fj = f):

I f 2

i

i f 2 sin Krij

Krijj

• Introducing the atomic-density distribution function ρi(rij) (for origin atom i) gives:

I f 2

i f 2 i rij sin Krij

Kriji dVi

where ∑j→∫.• Writing ρi(r) = <ρi(rij)>, and adding and subtracting a term in the macroscopic average atomic density, ρo, gives:

I f 2

i f 2 4r 2 r o sin Kr

Kri dr f 2 4r 2o sin Kr

Kri dr

where ∑i→N.

• The last term in ρo represents scattering due to the finite sample size (coinciding with the θ = 0o transmitted beam)

• Hence

I Nf 2 Nf 2 4r 2 r o sin KrKr

dr

• Defining the structure factor, S(K) as:

S K I K / Nf 2

and the reduced scattering function, F(K), as:

F(K ) K[S(K ) 1]then

F(K ) G(r )sin Kr drwhere the reduced RDF, G(r), is given by:

G(r ) 4r r o J(r )/ r 4ro

Nb Fourier transform

• F(K) and G(r) are therefore related via a Fourier transform

• Can obtain the RDF from measured scattering data I(K) → S(K) → F(K) by an inverse Fourier transform:

KKrKFrG dsin)(2)(

Nb limits: 0 < K < ∞