Ceramics Materials: Science & Engineering (세라믹스 ... - KOCW

Hyun-Suk Kim

Lecture 1:

Bulk Defect Chemistry 51570-00, Spring 2018

Ceramics Materials: Science & Engineering

(세라믹스 특론)

Defect Chemistry Review:

CLASSIFICATION DEFINITION EXAMPLES

VIBRATING ATOM TEMPORARY, SMALL DISPLACEMENT FROM IDEAL POSITION

ZERO POINT, THERMAL

ELECTRONIC CHARGE CHARGE CARRIER EXCITED FROM GROUND STATE BONDING

CONFIGURATION

ELECTRON (-), HOLE (+) EXCITON

CHEMICAL IMPURITY FOREIGN ATOM OF DIFFERING SIZE, VALENCE, ELECTRONEGATIVITY, AND/OR STRUCTURE RELATIVE TO HOST ATOMS/STRUCTURE

SUBSTITUTIONAL, INTERSTITIAL

POINT LATTICE DEFECT MISSING HOST ATOM EXTRA HOST ATOM

ATOM OCCUPYING WRONG LATTICE SITE

NON BRIDGING BOND

LATTICE VACANCY SELF INTERSTITIAL

ANTISITE DEFECT (COMPOUNDS)

(NONCRYSTALLINE MATERIALS) ONE DIMENSIONAL DEFECT ROW OF ATOMS AT EDGE OF EXTRA

HALF PLANE OF ATOMS

DISLOCATION (EDGE AND SCREW)

TWO DIMENSIONAL DEFECT BOUNDARY SEPARATING AN ERROR IN STACKING SEQUENCE OF ATOMIC PLANES

BOUNDARY BETWEEN TWO CRYSTALS OF DIFFERING RELATIVE ORIENTATION

STACKING FAULT

GRAIN BOUNDARY THREE DIMENSIONAL

DEFECT

MACROSCOPIC REGION OF DIFFERING DENSITY, CHEMICAL CONTENT,

COORDINATION ETC. FROM HOST

VOID FREE VOLUME

(NONCRYSTALLINE), DISORDER, DEFECT CLUSTER, PRECIPITATE

Imperfections in condensed matter

Defect Chemistry Review:

Why are defects important?

Defects have a profound impact on the various

properties of materials!

Defect Chemistry Review:

• Vacancies

• Interstitials

• Dopants / Impurities

What are the concentration of them?

Ionic charge carriers

Conductivity of system

Defect Thermodynamics:

Why is the perfect lattice not the most stable state?

Higher T  more defects Defects : Increase S

Increase H

Vacancies in a Metal:

→ ignore charge balance

no defects change due to introduction of defects

depends on the nature of the chemical bonding 0

for solids configurational

Also, electronic, vibrational, etc.

Vacancies in a Metal:

: # of ways arranging the system

distinguishable and energetically equivalent

𝑆 0, Ω 1

→ configurational entropy,

• Perfect lattice:

• S with defects: calculate  (per mole)

1) 1 vacancy in N_o atoms 2) 2 vacancies in N_o atoms

Ω 𝑁

Ω 𝑁 𝑁 1

2

account for

indistinguishable configs.

3) 3 vacancies in N_o atoms

Ω 𝑁 𝑁 1 𝑁 2

3 · 2

 n_v vacancies in N_o atoms

Ω 𝑁 𝑁 1 ⋯ 𝑁 𝑛 1 𝑛 !

𝑁 !

𝑛 ! 𝑁 𝑛 !

Vacancies in a Metal:

𝐺 𝐺 ∆ℎ 𝑛 𝑘𝑇𝑙𝑛 𝑁 !

𝑛 ! 𝑁 𝑛 !

Stirling’s approximation: 𝑙𝑛𝑋! 𝑋𝑙𝑛𝑋 𝑋

𝜕𝐺

𝜕𝑛 0 → 𝑛

𝑁 𝑛 𝑒𝑥𝑝 ∆ℎ 𝑘𝑇

𝑛

𝑁 𝑋

• Reintroduce other entropy terms

𝐺 𝐺 ∆ℎ 𝑛 𝑇∆𝑠 𝑛 𝑇𝑆

→ 𝑛

𝑁 𝑒𝑥𝑝 ∆ℎ 𝑇∆𝑠 𝑘𝑇

∆ ∆

Vacancies in a Metal:

1/T log

X V

Looks like a equilibrium constant of a chemical reaction

𝐾 𝑎

𝑎 𝑎 𝑒𝑥𝑝 ∆𝑔

𝑘𝑇

Ideal solution normal metal atoms

Defects in Ionic Compounds:

→ defect formation is constrained by charge balance

- Schottky defects:

Pairs of anion and cation vacancies

- Frenkel defects:

vacancy + interstitial

Cation Frenkel, F_C = cation vacancy + cation interstitial Anion Frenkel, F_A = anion vacancy + anion interstitial Typically, cation is smaller than anion in ionic solids.

Frenkel defects  F_C, “anti” Frenkel defects  F_A

e.g., NaCl

Na⁺

Cl^- Na⁺

Defects Notation: Kröger-Vink

→ specify species, effective charge and site

- D refers to an element or vacant site

- x represents the net effective charge with “” neutral,

“” negative, “” positive

- y refers to atomic sites in the structure

e.g., Na vacancy: -1 Cl vacancy: +1 Na interstitial: +1 Cl interstitial: -1

Ca²⁺ on a Na⁺ site: +1 e.g., Na, Cl, Ca (impurity or dopant), V (vacancy)

e.g., Na site, Cl site, Interstitial site

Defects in NaCl:

- “non-defects”:

- “vacancies”:

- “interstitial”:

- “impurities or dopants”:

·

·

Defects Concentration:

→ treat the generation of intrinsic defects as equilibrium rxns

- Schottky defects

𝑛𝑢𝑙𝑙 → 𝑉 𝑉

^·

𝑁𝑎 𝐶𝑙 → 𝑉 𝑉

^·

𝑁𝑎

_,

𝐶𝑙

_,

𝐾 𝑉 · 𝑉

^·

𝑒𝑥𝑝 ∆𝑔 𝑘𝑇

ideal solution behavior

→ charge balance

∴ 𝑉 𝑉

^·

𝐾

^/

𝑒𝑥𝑝

^∆

𝑒𝑥𝑝

^∆

𝑒𝑥𝑝

^∆

- Schottky equilibrium constant, K

_S

1/T log

[V]

Defects Reactions (In general):

→ must satisfy the following conditions

1)Mass balance:

2)Charge balance:

3)Site balance:

Mass must be conserved during the rxns.

The net charge of the system is conserved during the rxns.

The ratio of lattice site is fixed by the crystalline structure and does not change as a result of the rxns.

𝑛𝑢𝑙𝑙 → 𝑉 𝑉

^·

𝑁𝑎 → 𝑁𝑎

^·

𝑉

𝑛𝑢𝑙𝑙 → 2𝑉 3𝑉

^··

𝐴𝑙 → 𝐴𝑙

^···

𝑉

𝑂 → 𝑂 𝑉

^··

𝑛𝑢𝑙𝑙 → 𝑉 2𝑉 4𝑉

^··

e.g., NaCl e.g., Al

₂

O

₃

e.g., MgAl

₂

O

₄

Defects Reactions (In general):

- Schottky/Frenkel defects

- Extrinsic defects (doping/impurity)

- Electron-hole generation - Oxidation/reduction

e.g., Y₂O₃ in ZrO₂

• Substitutional incorporation

• Interstitial incorporation

𝑌 𝑂 2𝑌 3𝑂 𝑉

^··

𝑌 𝑂 → 2𝑌

^⋯

3𝑂

in principle, possible, but thermodynamically unfavorable

If within the solubility limit,

𝑌 ≡

Added amount Not fixed by

an equil. constant

Electron and Hole Generation:

·

𝐾 𝑛 𝑛 𝑁 𝑁 𝑒𝑥𝑝 𝐸 𝑘𝑇

density of states

→ Ionic defect can trap electrons and holes

𝐹𝑒 ℎ

^·

→ 𝐹𝑒

^·

e.g. 1

𝑉

^··

𝑒 → 𝑉

^·

𝑍𝑛

^··

ℎ

^·

→ 𝑍𝑛

^·

e.g. 2

transition metals

F-Center (or Color Center)

Oxidation/Reduction Rxns:

𝑂 1

2 𝑂 𝑔 𝑉

^··

2𝑒

1

2 𝑂 𝑔 𝑉

^··

𝑂 2ℎ

^·

𝐾 𝑛 𝑉

^··

𝑃 𝐾 𝑒𝑥𝑝 ∆𝑔 𝑘𝑇

𝐾 𝑝

𝑉

^··

𝑃

^/

𝐾 𝑒𝑥𝑝 ∆𝑔 𝑘𝑇

→ Equilibration of ionic solids with an ambient gas

(e.g., oxygen, halogen or metal vapor)

- Reducing conditions (low p

_O2

):

Extract oxygen from the structure

Electrons created upon reduction.

- Oxidizing conditions (high p

_O2

):

Incorporate oxygen into the structure

Holes created upon oxidation.

Oxidation/Reduction Rxns:

1

2 𝑂 𝑔 𝑉

^··

𝑂 2ℎ

^·

- Oxidizing conditions (high p

_O2

) (e.g., MgO):

1

2 𝑂 𝑔 𝑉

^··

2𝑒 𝑂 1

2 𝑂 𝑔 𝑂 𝑉 2ℎ

^·

1

2 𝑂 𝑔 2𝑒 𝑂 𝑉 1

2 𝑂 𝑔 𝑂 2ℎ

^·

1

2 𝑂 𝑔 2𝑒 𝑂

Alternative representations of the chemical process of oxidation!

It is convenient to choose just one representation that includes the prevailing defects in the system.

Oxidation/Reduction Rxns:

1

2 𝑂 𝑔 𝑉

^··

𝑂 2ℎ

^·

1

2 𝑂 𝑔 𝑉

^··

2𝑒 𝑂 1

2 𝑂 𝑔 𝑂 𝑉 2ℎ

^·

→ Only three out of four (equilibrium) defect reactions are independent !

-

2𝑒 2ℎ

^·

∴ 𝑛𝑢𝑙𝑙 𝑒 ℎ

^·

1

2 𝑂 𝑔 𝑉

^··

𝑂 2ℎ

^·

-

𝑉

^··

𝑉

∴ n𝑢𝑙𝑙 𝑉

^··

𝑉

𝑂 1

2 𝑂 𝑔 𝑉

^··

2𝑒 𝑛𝑢𝑙𝑙 𝑒 ℎ

^·

n𝑢𝑙𝑙 𝑉

^··

𝑉

2X

-

- ∴ 1

2 𝑂 𝑔 𝑉

^··

𝑂 2ℎ

^·

Extent of Nonstoichiometry:

- Stoichoimetric Compounds:

- Nonstoichoimetric Compounds

e.g., NaCl, MgO, Al₂O₃, Si₃N₄, etc

Satisfy Dalton’s Law of atoms combining in integer ratios.

e.g., TiO_2-x, SrTiO_3-x, Co_1-xO, etc

Stoichiometric compounds are special cases and exist in equilibrium at only specific temperatures and activities of constituents.

Generally the various atomic species in compounds are not found in integer ratios.

Examples (Nb 2 O 5 in TiO 2 ):

→ Electronic vs. Ionic Compensation of solutes

·

·

vs.

The prevailing compensation mechanism will depend on solute concentration, oxygen partial pressure, and temperature.