The Muppet’s Guide to:The Structure and Dynamics of Solids
7. Defects and Solutions
• Vacancy atoms• Interstitial atoms• Substitutional atoms
Point defects
Types of Imperfections
• Dislocations Line defects
• Grain Boundaries Area defects
Grain boundaries
D = b/
b
Internal surfaces of a single crystal where ideal domains (mosaic) meet with some misalignment: high-angle and small(low)-angle.
NB – in polycrystalline materials, grain boundaries are more extensive and may even separate different phases
Small-angle grain boundary equivalent to linear array of edge dislocations
bonding not fully satisfied region of higher energy, more reactive, impurities present.
(Callister: Materials Science and Engineering)
Point Defects
small substitutional atom
All of these defects disrupt the perfect arrangement of the surrounding atoms –relaxation effects
Schottky and Frenkel normally v low conc. since formation energy high
vacancy interstitial
large substitutional atom
Frenkel defect Schottky defect
Frenkel Defect• Tend to be found in ionic solids
with large size difference between the anion and cation
• The defect forms when an atom or cation leaves its place in the lattice, creating a vacancy, and becomes an interstitial.
• occur due to thermal vibrations
• occurrence depends on– size of ion– charge on ion– electronegativity– temperature
Ag
Ag
• Found in ionic crystals
• Oppositely charged ions leave their lattice sites, creating vacancies
• anion and cation vacancies balance such that charge neutrality is preserved
Schottky Defects
• Vacancies:
-vacant atomic sites in a structure.
• Self-Interstitials:
-"extra" atoms positioned between atomic sites.
Point Defects
Vacancydistortion of planes
self-interstitial
distortion of planes
(Callister: Materials Science and Engineering)
Material PropertiesDislocations & plastic deformation• Cubic & hexagonal metals - plastic deformation by
plastic shear or slip where one plane of atoms slides over adjacent plane by defect motion (dislocations).
• If dislocations don't move, deformation doesn't occur!
Adapted from Fig. 7.1, Callister 7e.
Edge Defect Motion
Dislocation Motion• Dislocation moves along slip plane in slip direction
perpendicular to dislocation line• Slip direction same direction as Burgers vector
Edge dislocation
Screw dislocation
Adapted from Fig. 7.2, Callister 7e.
(Callister: Materials Science and Engineering)
Dislocations & Materials Classes
• Covalent Ceramics
(Si, diamond): Motion hard. -directional (angular) bonding
• Ionic Ceramics (NaCl):
Motion hard. -need to avoid ++ and - - neighbours.
+ + + +
+++
+ + + +
- - -
----
- - -
• Metals: Disl. motion easier.
-non-directional bonding -close-packed directions for slip.
electron cloud ion cores
++
++
++++++++ + + + + +
+++++++
(Callister: Materials Science and Engineering)
Pinning dislocations
• dislocations make metals easier to deform• to improve strength of metals, need to stop dislocation motion
trap with:- impurity atoms;- other dislocations (work hardening;
- grain boundaries.
atom trap
(Callister: Materials Science and Engineering)
• Impurity atoms distort the lattice & generates stress.• Stress can produce a barrier to dislocation motion.
Modify Material Properties
• Smaller substitutional impurity
Impurity generates local stress at A and B that opposes dislocation motion to the right.
A
B
• Larger substitutional impurity
Impurity generates local stress at C and D that opposes dislocation motion to the right.
C
D
Increase material strength through substitution
(Callister: Materials Science and Engineering)
Modify Material Properties
• Grain boundaries are barriers to slip.• Barrier "strength" increases with Increasing angle of miss-orientation.• Smaller grain size: more barriers to slip.
Increase material strength through reducing Grain size
(Callister: Materials Science and Engineering)
Solid Solutions
Solid state mixture of one or more solutes in a solvent
Crystal structure remains unchanged on addition of the solute to the solvent
Mixture remains in a homogenous phase
Generally composed on metals close in the periodic tableNi/Cu, Pb/Sn etc.
Otherwise compounds tend to formNaCl, Fe2O3 etc.
Two outcomes if impurity (B) added to host (A):
• Solid solution of B in A (i.e., random dist. of point defects)
ORSubstitutional solid soln.
(e.g., Cu in Ni)Interstitial solid soln.
(e.g., C in Fe)
Point Defects in Alloys
(Callister: Materials Science and Engineering)
Hume-Rothery Rules – Substitutional Solutions
1. The solute and solvent should be of a similar size. (<15% difference)
2. The crystal structures must match.
3. Both solute and solvent should have similar electronegativity
4. The valence of the solvent and solute metals should be similar.
Rules to describe how an element might dissolve in a metal. Stable composition in equilibrium (thermodynamics)
Metals – Ni/Cu, Pd/Sn, Ag/Au, Mo/W
Phase Equilibria – Example
CrystalStructure
electroneg r (nm)
K BCC 0.93 0.235
Na BCC 1.00 0.191
• Both have the same crystal structure (BCC) and have similar electronegativities but different atomic radii.
• Rules suggest that NO solid solution will form.
K-Na
• K and Na sodium are not miscible.
Phase Equilibria – Example
CrystalStructure
electroneg r (nm)
Ni FCC 1.9 0.1246
Cu FCC 1.8 0.1278
• Both have the same crystal structure (FCC) and have similar electronegativities and atomic radii (W. Hume – Rothery rules) suggesting high mutual solubility.
Simple solution system (e.g., Ni-Cu solution)
• Ni and Cu are totally miscible in all proportions.
Hume-Rothery Rules – Interstitial Solution
1. The solute must be smaller than the interstitial sites in the solvent lattice
2. Solute and Solvent should have similar electro-negativities
Rules to describe how an element might dissolve in a metal. Stable composition in equilibrium (thermodynamics)
Light elements – H,C, N and O.
Phase Diagrams
• A phase diagram is a graphical representation of the different phases present in a material.
• Commonly presented as a function of composition and temperature or pressure and temperature
Applies to elements, molecules etc. and can also be used to show magnetic, and ferroelectric behaviour (field vs. temperature) as well as structural information.
• Components: The elements or compounds which are present in the mixture (e.g., Al and Cu)
• Phases: The physically and chemically distinct material regions that result (e.g., a and b).
Aluminum-CopperAlloy
Components and Phases
a (darker phase)
b (lighter phase)
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
Unary Phase DiagramsA pressure-temperature plot showing the different phases present in H2O.
Phase Boundaries
Upon crossing one of these boundaries the phase abruptly changes from one state to another. Latent heat not shown
Crossing any line
results in a structural
phase transition
Reading Unary Phase Diagrams
Melting Point (solid → liquid)
Boiling Point(liquid→ gas)
Sublimation (solid → gas)
As the pressure falls, the boiling point reduces, but the melting/freezing point remains reasonably constant.
Triple Point (solid + liquid + gas)
Reading Unary Phase Diagrams
Melting Point: 0°C Boiling Point: 100°C
Melting Point: 2°C Boiling Point: 68°C
P=1atm
P=0.1atm
Water Ice
http://images.jupiterimages.com/common/detail/13/41/23044113.jpg, http://www.homepages.ucl.ac.uk/~ucfbanf/ice_phase_diagram.jpg
• When we combine two elements... what equilibrium state do we get?• In particular, if we specify... --a composition (e.g., wt.% Cu – wt.% Ni), and --a temperature (T )
then... How many phases do we get? What is the composition of each phase? How much of each phase do we get?
Binary Phase DiagramsPhase BPhase A
Nickel atomCopper atom
Phase Equilibria: Solubility Limit– Solutions – solid solutions, single phase– Mixtures – more than one phase
• Solubility Limit: Max concentration for which only a single phase solution occurs.
Question: What is the solubility limit at 20°C?
Answer: 65 wt% sugar.
If Co < 65 wt% sugar: syrup
If Co > 65 wt% sugar: syrup + sugar.65
Sucrose/Water Phase Diagram
Pu
re
Su
gar
Tem
per
atu
re (
°C)
0 20 40 60 80 100Co =Composition (wt% sugar)
L (liquid solution
i.e., syrup)
Solubility Limit L
(liquid)
+ S
(solid sugar)20
40
60
80
100
Pu
re
Wat
er
Salt-Water(ice)
http://webserver.dmt.upm.es/~isidoro/bk3/c07sol/Solution%20properties_archivos/image001.gif