issues to address... what types of defects arise in solids? can the number and type of defects be...

ISSUES TO ADDRESS...

• What types of defects arise in solids?

• Can the number and type of defects be variedand controlled?

• How do defects affect material properties?

• Are defects undesirable?

• How do point defects in ceramics differ from thosein metals?

• In ceramics, how are impurities accommodatedin the lattice and how do they affect properties? 1

CHAPTER 5:IMPERFECTIONS IN SOLIDS

Imperfections in Solids

• Introduction and Motivation– Why study this?

• What I have shown you so far are “idealized” structures• You would get the impression that solids (particularly crystalline

solids) have “perfect” structures– This is clearly not the case

• All solids have imperfections, and these have profound influences on mechanical (e.g. alloys of Ag/Cu v pure silver), electrical (e.g. doping of silicon with boron) properties

• “Crystalline defect” – a lattice irregularity having one or more of its dimensions on the order of an atomic distance

• We will also organize this discussion around whether the defect involves a single atom (0 D), rows of atoms (1D), or interfacial (2D) defects

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• Vacancy atoms• Interstitial atoms• Substitutional atoms

• Dislocations

• Grain Boundaries

Point defects

Line defects

Area defects

TYPES OF IMPERFECTIONS


• Point defects in metals– Start simple: what happens when you have a crystal structure

and one of the atoms is missing vacancy– Every crystal has vacancies– Why? From a thermodynamic viewpoint, crystalline solids have

very low entropy (why?)– The introduction of vacancies increases the entropy of the crystal– Can estimate the number of vacancies in a crystal using a

Boltzmann type description

Tk

QNN

B

vv exp

Qv is the energy required for forming a vacancy

For a “typical” metal Nv/N slightly below the melting temperature is ~ 10-4

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• Vacancies:-vacant atomic sites in a structure.

Vacancydistortion of planes

• Self-Interstitials:-"extra" atoms positioned between atomic sites.

self-interstitialdistortion

of planes

POINT DEFECTS

Boltzmann's constant

(1.38 x 10-23 J/atom K)

(8.62 x 10-5 eV/atom K)

NDN

exp QDkT

No. of defects

No. of potential defect sites.

Activation energy

Temperature

Each lattice site is a potential vacancy site

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• Equilibrium concentration varies with temperature!

EQUIL. CONCENTRATION:POINT DEFECTS

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• We can get Q from an experiment.

NDN

exp QDkT

• Measure this... • Replot it...

1/T

N

NDln 1

-QD/k

slopeND

N

T

exponential dependence!

defect concentration

MEASURING ACTIVATION ENERGY

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• Find the equil. # of vacancies in 1m3 of Cu at 1000C.• Given:

ACu = 63.5g/mol = 8.4 g/cm3

QV = 0.9eV/atomNA = 6.02 x 1023 atoms/mole

8.62 x 10-5 eV/atom-K

0.9eV/atom

1273K

NDN

exp QDkT

For 1m3, N =NAACu

x x 1m3 = 8.0 x 1028 sites

= 2.7 · 10-4

• Answer:

ND = 2.7 · 10-4 · 8.0 x 1028 sites = 2.2x 1025 vacancies

ESTIMATING VACANCY CONC.

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• Low energy electron microscope view of a (110) surface of NiAl.• Increasing T causes surface island of atoms to grow.• Why? The equil. vacancy conc. increases via atom motion from the crystal to the surface, where they join the island.

Island grows/shrinks to maintain equil. vancancy conc. in the bulk.

Reprinted with permission from Nature (K.F. McCarty, J.A. Nobel, and N.C. Bartelt, "Vacancies inSolids and the Stability of Surface Morphology",Nature, Vol. 412, pp. 622-625 (2001). Image is5.75 m by 5.75 m.) Copyright (2001) Macmillan Publishers, Ltd.

OBSERVING EQUIL. VACANCY CONC.

Movie m3

http://www.nature.com/


• Point defects in ceramics– Point defects such as vacancies are also present in ceramics– Here it is a bit more involved, since ceramics have both cations

and anions– You might imagine that since the anions are (much?) larger than

the cations there are very few anion interstitial defects– “Defect structure” is often used to denote defects in ceramics –

again need to deal with charge neutrality

http://www.wiley.com/college/callister/0471470147/gallery/ch05/pages/Fig05_03.html


• Point defects in metals– A void left by an atom in the crystal structure is called a vacancy– An additional atom is called a self-interstitial

• This is an atom that is located in one of the interstitial voids in the crystal structure (remember those) that are usually not occupied

– These occur much less frequently than defects (why?)



• Point defects in ceramics– Because of charge neutrality, defects in ceramics must occur in

pairs– Frenkel defect -- cation-vacancy and cation-interstitial pair

(cation moves from regular site to interstitial site)– Schottky defect – cation-vacancy and anion-vacancy pair (move

one cation and one anion from the lattice to the external surface)


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• Frenkel Defect --a cation is out of place.

• Shottky Defect --a paired set of cation and anion vacancies.

Shottky Defect:

Frenkel Defect

• Equilibrium concentration of defects ~e QD /kT

Adapted from Fig. 13.20, Callister 5e. (Fig. 13.20 is from W.G. Moffatt, G.W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. 1, Structure, John Wiley and Sons, Inc., p. 78.) See Fig. 12.21, Callister 6e.

DEFECTS IN CERAMIC STRUCTURES


• Stoichiometry– Stoichiometric materials are those where the exact cation:anion

ratio is predicted from the chemical formula (i.e. NaCl); deviation from this leads to nonstoichiometric materials

– How would you get this?• Ions with multiple oxidation states (e.g. Fe+2, Fe+3)

• Impurities

These types of issues can lead todefects

Fe1-xOx < 1Non-stoichiometryDeficiency of Fe


• Impurities in solids (metals)– It is very hard to make ultra-high purity metals– Often turns out you don’t want to!– Mixtures of metals (or alloys) often have better properties than

the pure compounds – e.g. alloying silver with copper– Some definitions:

• Solid solution – single “solid phase” containing two (or more) metals– Formed upon addition of impurity

• For binary metal solid solutions – solvent/solute nomenclature used

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Two outcomes if impurity (B) added to host (A):• Solid solution of B in A (i.e., random dist. of point defects)

• Solid solution of B in A plus particles of a new phase (usually for a larger amount of B)

OR

Substitutional alloy(e.g., Cu in Ni)

Interstitial alloy

(e.g., C in Fe)

Second phase particle--different composition--often different structure.

POINT DEFECTS IN ALLOYS


• Solid solutions– In the case of a solid solution, upon addition of the impurity to the

host material, the crystal structure is maintained– Analogy to liquids

• Add two miscible liquids together (water & methanol)– The composition is homogeneous throughout

– Same idea with solid solutions – impurity atoms are randomly and uniformly dispersed within the solid

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• Low energy electron microscope view of a (111) surface of Cu.• Sn islands move along the surface and "alloy" the Cu with Sn atoms, to make "bronze".• The islands continually move into "unalloyed" regions and leave tiny bronze particles in their wake.• Eventually, the islands disappear.

Reprinted with permission from: A.K. Schmid, N.C. Bartelt, and R.Q. Hwang, "Alloying at Surfaces by the Migration of Reactive Two-Dimensional Islands", Science, Vol. 290, No. 5496, pp. 1561-64 (2000). Field of view is 1.5 m and the temperature is 290K.

ALLOYING A SURFACE

Movie m4


• Substitutional solid solutions– Regarding substitutional defects, there are several features

that determine how well the impurity dissolves in the host material

1. Atomic size factor – close in size, the better; if the difference is more than about +/-15%, form new solid phases

2. Crystal structure – crystal structures for both species should be the same (i.e. hard to dissolve an FCC metal in a BCC metal)

3. Electronegativity – closer the better; if the difference between the two is large, they will likely form an intermetallic compound instead of a substitutional solid solution

4. Valence – Metal will have a tendency to dissolve a metal of higher valency versus lower valency

– Example: Cu and Ni are “model” substitutional solid solution system• Size similar, both FCC, similar electronegativities, Ni+2, Cu+1


• Interstitial solid solutions– Impurity atoms fill voids or interstices among the host lattice– Given the high packing factors in metal structure (bcc, fcc, hcp),

the size of these positions is small

– So, dinterstitial atom < dhost atom

– Also very few of these defects (why?)– Example – Carbon forms an interstitial solid with iron (what is

this?) – however can only add 2% carbon


• Impurities in Ceramics– Can also have impurities in ceramics (again, issue of anions v

cations, electroneutrality)

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• Impurities must also satisfy charge balance

• Ex: NaCl Na+ Cl-

• Substitutional cation impurity

• Substitutional anion impurity

initial geometry Ca2+ impurity resulting geometry

Ca2+

Na+

Na+Ca2+

cation vacancy

initial geometry O2- impurity

O2-

Cl-

anion vacancy

Cl-

resulting geometry

IMPURITIES


• Sections 5.5, 5.6 – Read– 5.5 – can have point defects in polymers – 5.6 – composition units conversion, you should know all of this already

from 204(?)

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Definition: Amount of impurity (B) and host (A) in the system.

• Weight %

Two descriptions:• Atom %

CB = mass of Btotal mass

x 100 C'B = # atoms of Btotal # atoms

x 100

• Conversion between wt % and at% in an A-B alloy:

CB = C'BAB

C'AAA + C'BABx 100 C'B =

CB/AB

CA/AA + CB/AB

• Basis for conversion:

mass of B = moles of B x AB

atomic weight of B

mass of A = moles of A x AA

atomic weight of A

COMPOSITION


• Dislocations/linear defects– A dislocation is a linear (one-dimensional) defect around which

atoms are misaligned– Simplest type of dislocation – the presence of an extra portion of

a plane of atoms, which terminates inside the crystal • This is called an edge dislocation

This is an extra plane of atoms

Note lattice distortion around plane

Atoms above dislocation line are squeezed together, those below are pushed apart

Magnitude and direction oflattice distortion is expressedin terms of the Burgers vector


• Edge dislocation– Few other points– Where the plane ends is often called the “edge dislocation line”

denoted by the ┴ symbol– The symbol points up if the extra plane is in the “top” of the

crystal; points down if it is the “bottom” of the crystal

Also note the Burgers vector b

This is perpendicular to the edge dislocation

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• are line defects,• cause slip between crystal plane when they move,• produce permanent (plastic) deformation.

Dislocations:

Schematic of a Zinc (HCP):• before deformation • after tensile elongation

slip steps

LINE DEFECTS


• Screw dislocation– Can think of this as being due to a

shear stress applied to the crystal– The upper front region of the crystal

is shifted one atomic distance to the right

– This also can be thought of as a linear dislocation along the line AB in the bottom Figure


• Screw dislocation– Atomic positions above the “slip

plane” are open circles, those below are solid circles

– Here the Burgers vector b is parallel to the screw dislocation

In reality, most defects have components of both types … these are referred to as mixed dislocations

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• Dislocations slip planes incrementally...• The dislocation line (the moving red dot)... ...separates slipped material on the left from unslipped material on the right.

Simulation of dislocationmotion from left to rightas a crystal is sheared.

(Courtesy P.M. Anderson)

INCREMENTAL SLIP

Click on image to animate

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• Dislocation motion requires the successive bumping of a half plane of atoms (from left to right here).• Bonds across the slipping planes are broken and remade in succession.

Atomic view of edgedislocation motion fromleft to right as a crystalis sheared.

(Courtesy P.M. Anderson)

BOND BREAKING AND REMAKING

Click on image to animate


• Interfacial defects– We have now talked about zero-dimensional (i.e. point) and one-

dimensional (linear) defects. Can also have two-dimensional defects

– These are boundaries that normally separate regions of the material with different crystal structures and/or crystallographic orientation (remember about grains?)

– The most common – external surfaces, grain boundaries, stacking faults, twin boundaries, and phase boundaries


• Interfacial defects– External surfaces

• Why defects? Surface atoms are not bonded to the same number of atoms as the “bulk” atoms

• They are therefore in a higher energy state

• Nature (thermodynamics) says minimize your free energy (here dominated by surface free energy effects)

• In liquids – typically get spherical droplets (why?)– Solids can’t do this

• However, it is common to see structural rearrangement at surfaces to minimize energy


• Interfacial defects– Grain boundaries

• Boundaries between domains (or grains) of a polycrystalline material that possesses different crystallographic orientations

• Within the boundary there is mismatch of the atomic arrangements as compared to the crystallographic domains

• Orientation mismatch can be minor (small- or low-angle grain boundaries) or major (large- or high-angle grain boundaries)

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Grain boundaries: • are boundaries between crystals. • are produced by the solidification process, for example. • have a change in crystal orientation across them. • impede dislocation motion.

grain boundaries

heat flow

Schematic

Adapted from Fig. 4.7, Callister 6e. Adapted from Fig. 4.10, Callister 6e. (Fig. 4.10 is from Metals Handbook, Vol. 9, 9th edition, Metallography and Microstructures, Am. Society for Metals, Metals Park, OH, 1985.)

~ 8cmMetal Ingot

AREA DEFECTS: GRAIN BOUNDARIES



• Tilt boundary – series of edge dislocations leading to a small-angle grain boundary

Key point here – the edge dislocations are aligned with respect to one another



• Turns out that atoms along the grain boundary have less regular bonding as compared to that of the “bulk”

– This is analogous to surface atoms

• As a result they have a higher energy

• Tend to be more reactive

• Often impurity atoms will segregate into/along the grain boundaries

• Grow at elevated temperatures to reduce the total boundary energy


• Interfacial defects– Twin boundaries

• Special type of grain boundary – across the grain boundary there is a mirror symmetry

The region between the boundary is called a twin

These are formed due to applied shear forces (mechanical twins) or during high temperature heat treatments (annealing twins)

These twins occur on a definite crystallographic plane and in a specific direction


• Interfacial defects– Stacking faults

• Exactly what it sounds like … the layers in a structure are stacked incorrectly

• For example, in the FCC structure an interruption of the ABCABC.. stacking sequence of close-packed planes

– Phase boundaries – domains between solids that phase separated (multiphase materials)


• Bulk/volume defects– These are “3-D” defects– Pores – holes in the material– Cracks– “Foreign inclusions” – a large impurity domain?– Usually result from processing/fabrication– All effect material properties

• Atomic vibrations – read• Microscopic examination – read

– Key point – microscopy allows you to generate real space images of solids

– Resolution of the image (i.e. size of the objects/features you can see) depends on the wavelength of the radiation used

Cu atomic shells for (Cu0.25Ni0.75)343 at various temperatures during the heating process (MD simulations, S-P Huang and PB Balbuena, J. Phys. Chem. B, 2002)

0K 200 K 300 K

400 K 700 K 1300 K

Cu0.25Ni0.75 nanocluster (343 atoms) deposited on graphite (not shown)

700 K 800 K

900 K 1200 K

(S-P Huang, D. S. Mainardi and P. B. Balbuena, Surf. Sci., 2003)

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• Useful up to 2000X magnification.• Polishing removes surface features (e.g., scratches)• Etching changes reflectance, depending on crystal orientation.

microscope

close-packed planes

micrograph ofBrass (Cu and Zn)

Adapted from Fig. 4.11(b) and (c), Callister 6e. (Fig. 4.11(c) is courtesyof J.E. Burke, General Electric Co.

0.75mm

OPTICAL MICROSCOPY (1)

Fe-Cr alloy

microscope

grain boundarysurface groove

polished surface

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Grain boundaries...• are imperfections,• are more susceptible to etching,• may be revealed as dark lines,• change direction in a polycrystal.

Adapted from Fig. 4.12(a) and (b), Callister 6e.(Fig. 4.12(b) is courtesyof L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].)

ASTM grain size number

N = 2n-1

no. grains/in2 at 100x magnification

OPTICAL MICROSCOPY (2)

Electron Microscopy

• Instead of light radiation, uses beams of electrons

• A high-velocity electron becomes wave-like (QM), and its wavelength is inversely proportional to its velocity. If accelerated (high V), wavelengths could be of ~0.003 nm (3pm). => high magnification and high resolution

Electron microscopy

• Transmission electron microscope (TEM): an electron beam goes through an specimen (usually prepared as a thin foil) => transmission of a high % of the incident beam.

• Transmitted beam is projected for visualization. Magnifications of 1,000,000X or more. (useful for dislocations)

Electron microscopy

• Scanning electron microscope (SEM): the surface of a specimen is scanned with an electron beam, and the reflected beam of electrons is collected on a cathode ray tube (similar to a TV screen). The surface must be electrically conductive. Magnifications from 10 to 50,000X.

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• Atoms can be arranged and imaged!

Carbon monoxide molecules

arranged on a platinum (111)

surface.

Photos produced from the work of C.P. Lutz, Zeppenfeld, and D.M. Eigler. Reprinted with permission from International Business Machines Corporation, copyright 1995.

Iron atoms arranged on a copper (111)

surface. These Kanji characters

represent the word “atom”.

SCANNING TUNNELING MICROSCOPY

Uses a tiny probe with a very sharp tip brought in very close proximity of the specimen. The probe movements are monitored, and a 3-D image of the surface is generated.

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• Point, Line, and Area defects arise in solids.

• The number and type of defects can be varied and controlled (e.g., T controls vacancy conc.)

• Defects affect material properties (e.g., grain boundaries control crystal slip).

• Defects may be desirable or undesirable (e.g., dislocations may be good or bad, depending on whether plastic deformation is desirable or not.)

SUMMARY

Reading: Chapter 5

HW # 4: Due Monday, February 19th

5.2; 5.7; 5.9; 5.11; 5.27; 5.31; 5.32; 5.34; 5D2

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issues to address... what types of defects arise in solids? can the number and type of defects be...

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