Crystal Imperfections in Solids [7]

1>

Agregation of atoms (temperature decreasing):vapour/gas liquid solidSolidification: result of casting of molten material

• Step 1: Nuclei formation• Step 2: Nuclei grow to form crystals – grain

structure

Schematic illustration of the stages during solidification of molten metal; each small square represents a unit cell. (a) Nucleation of crystals at random sites in the molten metal; note that the crystallographic orientation of each site is different. (b) and (c) Growth of crystals as solidification continues. (d) Solidified metal, showing individual grains and grain boundaries; note the different angles at which neighboring grains meet each other.

Strength of Materials

2>

Based on the bond strength most materials should be much stronger than they are.

The strength for a strong ionic bond should be about 7 GPa …

… but more typical strength is 300 MPa !

Why do we have three orders of magnitude difference?

Answer: Imperfections/defects in structure

Crystal Imperfections in Solids

3>

The atomic arrangements in a crystalline lattice is almost always not perfect.

There are “defects” in the way atoms are arranged in the crystalline solids.

So we can say that in crystalline solids some Lattice Irregularities are always present.

These crystalline defects are not bad. Some are intentionally introduced to improve the material.

Types of Crystalline Defects

4>

CRYSTALLINE DEFECTS: classified on the basis of their geometry and dimensionallity.

POINT DEFECTS (no dimensional)

LINE DEFECTS (1D)

INTERFACIAL/SURFACE DEFECTS (2D)

VOLUME DEFECTS (3D)

Classification of Defects Based on Dimensionality

5>

0D(Point defects)

1D(Line defects)

2D(Surface / Interface)

3D(Volume defects)

Vacancy

Impurity

Frenkeldefect

Schottkydefect

Dislocation Twins

Precipitate

Voids / Cracks

Surface

Interphaseboundary

Grainboundary

Twinboundary

Stackingfaults

Dimensional scale of defects

6>

significant effect on mechanical properties

Point Defects

7>

POINT DEFECT: Imperfect point-like regions in the crystal about the size of 1-2 atomic diameters

0D(Point defects)

Vacancy

Impurity

Frenkel defect

Schottky defect

Non-ioniccrystals

Ioniccrystals

Interstitial

Substitutional

Point Defects

8>

Vacancy: missing atom from an atomic siteAtoms around the vacancy displacedTensile stress field produced in the vicinity

Tensile stress field of vacancy

X

Vacancies

9>

Usually introduced during solidification, at high TThe vacancy concentration in pure elements is very low at low temperatures.

The probability that an atomic site is vacancy ~ 10-6

at low temperatureThe probability that an atomic site is vacancy ~ 10-3

at melting temperaturecan affect physical and electronic structures around them → influence properties like color, conductivitythey play a critical role in diffusion: control the self diffusion and substitutional diffusion ratesMovements of atoms coupled with movements of vacancies

Vacancies Equilibrium of Point Defects

10>

Each lattice site is a potential vacancy site, so the equilibrium number of vacancies formed as a result of thermal vibrations may be calculated from thermodynamics.At equilibrium, the fraction of lattices that are vacant (or vacancy concentration) at a given temperature is given approximately by the equation:

where n is the number of point defects (number of vacancy sites) in N sites and Gf is free energy of formation of the defects (vacancy), T is the absolute temperature, k is the Boltzman constant (1.38 x 10-23 J/atom.K or 8.62 x 10-5 eV/atom.K).A large numbers of additional (non equilibrium) vacancies can be introduced by thermomechanical processing.

( )kTG feNn /−=

The Effect of Temperature on Vacancy Concentration

11>38

322

/cmvacancies10815.1

298KKmol

cal1.987

molcal20,000

exp . cm

atoms1047.8

exp

×=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

×−

−⎟⎠⎞

⎜⎝⎛ ×=

⎟⎠⎞

⎜⎝⎛=

RTnn Qν

ν

Calculate the concentration of vacancies in FCC copper at room temperature (25oC). Assume that 20,000 cal are required to produce a mole of vacancies in copper.Solution: The lattice parameter of FCC copper is 0.36151 nm. The number of copper atoms, or lattice points, per cm3 is:

3 2238 atoms/cmcopper 1047.8

)cm106151.3(atoms/cell 4

×=×

= −n

At room temperature, T = 25 + 273 = 298 K

Impurities in Solids

12>

Pure metal, containing only one type of atoms, is not possible.Impurity atoms are always present.These atoms exists as point defects.In alloys, impurity atoms (alloying element atoms) are intentionally added. An alloy is usually a solid solution of two or more types of atoms.Examples: Fe + C = steel

Cu + Zn = brassNi + Cu = monelAl + Cu = dural

Types of Solid Solutions

13>

SUBSTITUTIONAL IMPURITY• Foreign atom replacing the parent atom in the crystal• E.g. Cu sitting in the lattice site of FCC-Ni

INTERSTITIAL IMPURITY• Foreign atom sitting in the void of a crystal• E.g. C sitting in the octahedral void in HT FCC-Fe

Tensile Stress Fields Compressive Stress Fields

INTERSTITIALSUBSTITUTIONAL

Hume-Rothery Rules for Solid Solution

14>

Hume-Rothery (1899-1968) states 4 Conditions that must be satisfied for obtaining large solubility of the substitutional solute in a given solvent lattice:

1. Atomic Size Factor: The atomic size difference between the solute and solvent atoms must be less than ± 15%.

2. Crystal Structure: Crystal structure of both solute and solvent must be same.

3. Electronegative: The difference must be small. If this difference is large ionic compound will form instead of solid solution.

4. Valence: Higher valence metals will dissolve easily than low valence metals.When one or more is violated, partial solubility is obtained.

Ni + Cu: will they have large Solid Solubility?

Checking 4 H-R rules …

Hume-Rothery Rules for Solid Solution

15>

+1+2Valence1.91.8Electronegativity

FCCFCCCrystal structure0.128nm0.125nmAtomic Size

CuNirule

Answer: Yes, they will total solubility.

Zn + Cu: will they have large Solid Solubility?

Checking 4 H-R rules …

Hume-Rothery Rules for Solid Solution

16>Answer: No, they will not (partial solubility).

+1+2Valence1.91.6Electronegativity

FCCHCPCrystal structure0.128nm0.133nmAtomic Size

CuZnrule

Carbon Solid Solution in Iron

17>

In FCC iron, carbon atoms are located at octahedralsites. In BCC iron, carbon atoms enter tetrahedralsites which are larger than octahedral ones. What would be the atomic percentage of carbon in each type of iron if all the interstitial sites were filled?SOLUTION: We can find a total of 12 tetrahedral interstitial sites for

BCC structure. If all sites of BCC cell are occupied atomic percentage of carbon in BCC iron would be:

The number of octahedral interstitial sites is 4 interstitial sites per FCC unit cell. So, the atomic percentage of carbon in FCC iron would be:

Carbon Solid Solution in Iron

18>

Interstitial C sitting in the octahedral void in HT FCC-Fe

rOctahedral void / rFCC atom = 0.414rFe-FCC = 1.29 Å ⇒ rOctahedral void = 0.414 x 1.29 = 0.53 ÅrC = 0.71 Å⇒ Compressive strains around the C atomSolubility limited to 2 wt% (9.3 at%)

Interstitial C sitting in the tetrahedral void in RT BCC-Fe

rTetrahedral void / rBCC atom = 0.29 • rC = 0.71 ÅrFe-BCC = 1.258 Å ⇒ rTetrahedral void = 0.29 x 1.258 = 0.364 Å► But C sits in smaller octahedral void - displaces fewer atoms⇒ Severe compressive strains around the C atomSolubility limited to 0.008 wt% (0.037 at%)

Specification of Composition or Concentration

19>

m = mass; n = moles (atoms) - Compositions are easily converted from one type to the other by manipulating m to n, or vice versa, using the atomic weight, “A”.

1

11 A

mnm =being

– WEIGHT PERCENT 100x 21

11 mm

mC+

=

100x 21

1'1

mm

m

nnnC+

=– ATOM PERCENT

(wt%)

(at%)

Specification of Composition or Concentration

20>

Example: Determine the composition, in atom percent, of an alloy that consists of 97 wt% aluminum and 3 wt% copper.

Cu1.3%at C - 100 C Al98.7%at 0.047 3.595

3.595

Cu mol 0.047 Cu) g/mol (63.55

Cu) (3gatoms)Cu of(number

Al mol 3.595 Al) g/mol (26.98

Al) (97gatoms) Al of(number

Al) of weight (atomicatoms) Al of (massatoms) Al of(number

atoms)Cu of(number atoms) Al of(number atoms) Al of(number

AlCu ===+

=

==

==

=

+=

Al

Al

C

C

Solid Solution Strengthening in Steels

21>

Interstitial solute

Substitutionalsolute

Point Defects in Ceramics

22>

Vacancies-- vacancies exist in ceramics for both cations and anions Interstitials-- interstitials exist for cations-- interstitials are not normally observed for anions because

anions are large relative to the interstitial sites

Adapted from Fig. 5.2, Callister & Rethwisch 3e.(Fig. 5.2 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.)

Cation Interstitial

Cation Vacancy

Anion Vacancy

Point Defects in Ceramics

23>

Frenkel DefectTo maintain the charge neutrality, a cation vacancy-cationinterstitial pair occur together. The cation leaves its normal position and moves to the interstitial site.

Point Defects in Ceramics

24>

Schottky DefectTo maintain the charge neutrality, remove 1 cation and 1 anion; this creates 2 vacancies.

Point Defects in Ceramics

25>

Since there are both anions and cations in ceramics, a substitutionalimpurity will replace the host ion most similar in terms of charge.

Charge balance must be maintained when impurities are present.• Example: NaCl Na + Cl -

• Substitutional cation impurity

without impurity Ca2+ impurity with impurity

Ca2+

Na+

Na+Ca2+

cationvacancy

• Substitutional anion impurity

without impurity O2- impurity

O2-

Cl-

anion vacancy

Cl-

with impurity

Semiconductor Ceramics

26>

Intrinsic semiconductor - A semiconductor in which properties are controlled by the element or compound that makes the semiconductor and not by dopants or impurities.

Extrinsic semiconductor - A semiconductor prepared by adding dopants, which determine the number and type of charge carriers.

Doping - The addition of a small percentage of foreign atoms in the regular crystal lattice of silicon or germanium produces dramatic changes in their electrical properties, producing n-type and p-type semiconductors.

Extrinsic Semiconductor

27>

Semiconductor N-type: The addition of penta-valent impurities such as Antimonium, Arsenic or Phosphorus contribute free electrons, greatly increasing the conductivity of the intrinsic semiconductor.

Donor impurities:SbAsP

Extrinsic Semiconductor

28>

Semiconductor P-type: The addition of trivalentimpurities such as Boron, Aluminium or Gallium to an intrinsic semiconductor creates deficiencies of valence electrons, called "holes".

Acceptor impurities:BAlGa

Point Defects

29>

Linear Defects

30>

Dislocations are linear defects and represent a line around which atoms in the crystalline lattice are misaligned.

Types of Dislocations:EDGE DISLOCATION: A dislocation introduced into the crystal by adding an ‘‘extra half plane’’ of atoms.SCREW DISLOCATION: A dislocation produced by skewing a crystal so that one atomic plane produces a spiral ramp about the dislocation.“MIXED” DISLOCATION: A dislocation that contains partly edge components and partly screw components

Edge Dislocation

31>Compressive stress field is colored RED and tensile stress field is BLUE.

Burgers vector b ⊥ dislocation lineb

Shear stress

Screw Dislocation

32>

There are not compressive or tensile stress fields associated to screwdislocation, only pure shear around dislocation line.

Dislocationline Burgers

vector b

Burgers vector b || dislocation line

bTop view

Shear stress

Edge, Screw, and Mixed Dislocations

33>

Screw

Edge

Mixed

During sliding Burgers vector does not change.

Burgers Vector

34>

Burgers Vector b represents the magnitude and direction of lattice distortion created by the dislocation.

– EDGE DISLOCATION b is perpendicular to dislocation – SCREW DISLOCATION b is parallel to dislocation

Slip Systems

35>

Dislocations move more readily in some crystal planes and directions than in others as we will see.The combination of both the slip plane and direction form the slip system.Slip plane is generally taken as the closest packed plane in the system. Slip direction is taken as the direction on the slip plane with the highest linear density.The line direction of a screw dislocation is in the same direction as its Burger’s vector.An edge dislocation has its Burger’s vector perpendicular to the line direction of a dislocationA dislocation having a line direction not parallel or perpendicular to the Burger’s vector is considered a mixed dislocation.

Schematic of Dislocation Slip

36>

Edge dislocation Screw dislocation

Slip Systems for Dislocations

37>

Examples of slip planeand direction for BCC

Burgers Vector Calculation

38>

Calculate the length of the Burgers vector in copper.

The length of the Burgers vector, or the repeat distance, is:

b = 1/2(0.51125 nm) = 0.25563 nm

SOLUTION: Copper has an FCC crystal structure. The lattice parameter of copper (Cu) is 0.36151 nm. The close-packed directions, or the directions of the Burgers vector, are of the form <110>. The repeat distance along the directions is one-half the face diagonal, since lattice points are located at corners and centers of faces.

Formation of Dislocations

39>

Frank-Read sources:

Dislocations propagatingon slip plane from a Frank-Read source

Two atractiveFrank-Read interacting

Two repulsive Frank-Read interacting

http://www.numodis.fr/tridis/

Dislocation Motion and Interaction

40>dislocations moving through a forest of dislocations

http

://w

ww

.num

odis

.fr/tr

idis

/

Dislocation Motion and Interaction

41>pile-up at a grain boundary and overcoming it.

http

://w

ww

.num

odis

.fr/tr

idis

/

Dislocation Motion and Interaction

42>dislocation multiplication and transmission in a bicrystal

http

://w

ww

.num

odis

.fr/tr

idis

/

Dislocation Motion and Interaction

43>

dislocation bowing a spherical particle.

http

://w

ww

.num

odis

.fr/tr

idis

/

Dislocation Motion and Interaction

44>

Mobility of an edge dislocation with different kinds of solute atoms

Observing Dislocations in Thin Foils

45>

(c) 2003 Brooks/C

ole Publishing / Thomson Learning

Dislocations can be observed in thin foil samples,a very thin piece of material (< 100nm), whichbe able to diffract a high voltage accelerated electronbeam of a transmission electron microscope (TEM).

top view side view TEM micrograph

Observing Dislocations in Thin Foils

46>

Source: http://www.numodis.fr/tridis/TEM/index.html

Interfacial Defects

47>

INTERFACIAL DEFECTS are boundaries that have 2 or 3 dimensions and normally separate regions of the materials that have different crystal structures and/or crystallographic orientations.

EXTERNAL SURFACES: One of the most obvious imperfection boundaries is the external surface because the crystal structure terminates.

Surface atoms are not bonded to the maximum number of nearest neighbors so they have higher energy state than interior atoms.INTERNAL SURFACES:

grain boundariestwin boundariesStacking faults

Of all these, the grain boundaries are the most important from the mechanical properties point of view.

Grain Structure

48>

Crystalline solids generally consist of millions of individual grains separated by boundaries. Each grain is a single crystal. Within each individual grain there is a systematic packing of atoms. Therefore each grain has different orientation and is separated from the neighboring grain by grain boundary.

grain

grain boundary

Polycrystalline Materials

49>

At the grain boundary, there is a disturbance in theatomic packing.

HAGB: θ > 15 degrees

Hall-Petch mechanism: dependence of strength on grainsize for a number of metals and alloys.

Polycrystalline Materials

50>

HALL-PETCH: Small grain size

equates to more strength

Grain Growth: Simulations in 2D & 3D

51>https://www.youtube.com/watch?v=J_2FdkRqmCAhttps://www.youtube.com/watch?v=Ac_ca_NeRnw

Tilt Boundary

52>

The simplest grain boundary consists of a configuration of edge dislocations between two grains.

The misfit in the orientation of the two grains (one on each side of the boundary) is accommodated by a small perturbation of the regular arrangement of crystals in the boundary region.

Schematic shows some vertical atomic planes termination in the boundary and each termination is represented by an edge dislocation.

Twist Boundary

53>

When the angle of misorientation is parallel to the boundary due to an array of screw dislocations.

Twin Boundary

54>

A twin boundary is a special type of grain boundary across which there is a specific mirror lattice symmetry. Twin results from atomic displacement that are produced from applied mechanical shear forces, and also during annealing heat treatments following deformation (annealing twins).

Twinning (FCC): {111} <211>

http://practicalmaintenance.net/?p=1135

Comparison

55>

Slip Twinning

orientation of atomsremains the same

reorientation of atomicdirection across twin plane

displacements take placein exact atomic spacings

atomic displacement is lessthan interatomic spacing

Stacking Faults

56>

Error in packing sequence:

SF

perfect defect

Stacking Faults

57>

Bright Field Image in TEM:

http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_6/backbone/r6_3_3.html

Bulk or Volume Defects

58>

Much larger defects than the previous ones, usually introduced during processing and fabrication steps

Examples include:– Pores– Cracks– Foreign Inclusions– Other Phases

Stress-corrosion cracking of na Inconel heat exchanger tube.

CALLISTER JR, W. D. AND RETHWISCH, D. G. Materials Science and Engineering: An Introduction, 9th edition.

John Wiley & Sons, Inc. 2014, 988p. ISBN: 978-1-118-32457-8.ASHBY, M. and JONES, D. R. H.

Engineering Materials 1: An Introduction to Properties, Applications and Design. 4th Edition. Elsevier Ltd. 2012, 472p. ISBN 978-0-08-096665-6.

CALLISTER JR, W. D. AND RETHWISCH, D. G. Fundamentals of Materials Science and Engineering: An Integrated Approach, 4th ed.

John Wiley & Sons, Inc. 2012, 910p. ISBN 978-1-118-06160-2.MITTEMEIJER, E. J.

Fundamentals of Materials Science: The Microstructure–Property Relationship Using Metals as Model Systems. Springer-Verlag Berlin. 2010, 594p. ISBN 978-3-642-10499-2.

ASKELAND, D. AND FULAY, P. Essentials of Materials Science & Engineering, 2nd Edition.

Cengage Learning. 2009, 604p. ISBN 978-0-495-24446-2.ABBASCHIAN, R., ABBASCHIAN, L., AND REED-HILL, R. E.

Physical Metallurgy Principles, 4th Ed. Cengage. 2009, 750p. ISBN 978-0-495-08254-5.SMALLMAN, R. E. and NGAN, A.H.W.

Physical Metallurgy and Advanced Materials, 7th Edition. Elsevier Ltd. 2007, 650p. ISBN 978-0-7506-6906-1.

http://www.tf.uni-kiel.de/matwis/amat/def_en/

References

59Nota de aula preparada pelo Prof. Juno Gallego para a disciplina Ciência dos Materiais de Engenharia.® 2015. Permitida a impressão e divulgação. http://www.feis.unesp.br/#!/departamentos/engenharia-mecanica/grupos/maprotec/educacional/