Enggineering Material Coordination No

8/10/2019 Enggineering Material Coordination No

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Atomic Packing Factor

Atomic Packing Factor (APF) is defined as the

volume of atoms within the unit cell divided by

the volume of the unit cell.


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Crystal Structure 3

1-CUBIC CRYSTAL SYSTEM

In the unit cell on the left, the atoms at the corners are cut

because only a portion (in this case 1/8) belongs to that cell.

The rest of the atom belongs to neighboring cells.

Coordinatination number of simple cubic is 6.

a- Simple Cubic (SC)

a

bc
http://www.kings.edu/~chemlab/vrml/simcubun.wrl


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a- Simple Cubic (SC)


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Crystal Structure 6

Atomic Packing Factor of SC


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Crystal Structure 7

b-Body Centered Cubic (BCC)

BCC has eight nearest neighbors.

Each atom is in contact with itsneighbors only along the body-

diagonal directions.

Many metals (Fe,Li,Na..etc),

including the alkalis and severaltransition elements choose the

BCC structure.

a

b c


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+Body Centred Cubic (BCC) Lattice Sphere Motif

=

Body Centred Cubic Crystal

Unit cell of the BCC lattice

Atom at (, , )

Atom at (0, 0, 0)

Space filling model

Central atom is coloured differently for better visibility


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Crystal Structure 9

0.68=V

V=APF

cellunit

atomsBCC

2 (0,433a)

Atomic Packing Factor of BCC


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Crystal Structure 10

c- Face Centered Cubic (FCC)

There are atoms at the corners of the unit cell and at the center ofeach face.

Face centered cubic has 4 atoms so its non primitive cell.

Many of common metals (Cu,Ni,Pb..etc) crystallize in FCCstructure.


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+Face Centred Cubic (FCC) Lattice Sphere Motif

=

Cubic Close Packed Crystal

(Sometimes casually called the FCC crystal)

Atom at (, , 0)

Atom at (0, 0, 0)

Unit cell of the FCC lattice

Space filling model


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4 (0,353a)

0.68=V

V=APF

cellunit

atomsBCCFCC

0,74

Atomic Packing Factor of FCC


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2 - HEXAGONAL SYSTEM

A crystal system in which three equal coplanar axes

intersect at an angle of 60 , and a perpendicular to the

others, is of a different length.


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Both FCC and HCP structures are close packedAPF = 0.74.


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VOLUME, PLANAR, AND LINEAR DENSITY

Volume density --

Planar density --

Linear Atomic density --

celltvolume/uni

cellmass/unit==metalofdensityVolume

v

planeofareaselected

dintersectecentersatom#==densityatomicPlanar

p

lineoflengthselected

dintersectediametersatom#==densityatomicLinear

l


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BCC FCC HCP


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LINEAR & PLANAR DENSITIES

Linear density (LD) = number ofatoms centered on a direction vector/ length of direction vector LD (110) = 2 atoms/(4R) = 1/(2R)

Planar density (PD) = number ofatoms centered on a plane / area ofplane PD (110) = 2 atoms / [(4R)(2R2)] =

2 atoms / (8R22) = 1/(4R22) LD and PD are important

considerations during deformationand slip; planes tend to slip or

slide along planes with high PDalong directions with high LD


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THEORETICAL DENSITY,

Density = mass/volume

mass = number of atoms per unit cell * mass of each atom

mass of each atom = atomic weight/avogadros number


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Why?

Metalshave... close-packing

(metallic bonding)

large atomic mass

Ceramicshave... less dense packing

(covalent bonding)

often lighter elements

Polymershave... poor packing

(often amorphous)

lighter elements (C,H,O)

Compositeshave... intermediate values Data from Table B1, Callister 6e.

DENSITIES OF MATERIAL CLASSES

metals> ceramic s> polymer s


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Someengineering applications require single crystals:

--diamond single

crystals for abrasives--turbine blades

Single crystal: when the periodic and repeated arrangement

of atoms is perfect and extends throughout the entirety of the

specimen


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POLYCRYSTALLINE MATERIALS

Nuclei form during solidification, each of which grows into crystals


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No crystal has 100% flawless structure


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Crystal defects

Defects can affect

Strength

Conductivity

Deformation style

Color


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Crystal DefectsSteel spheres:

a) Regular packed array with 3 pointdefects

b) Pointand line defects

c) Mosaic(or domains) separated by

defect boundaries

Fig 3.50 of Klein

and Hurlbut, Manual

of Mineralogy,

John Wiley and Sons


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Point defects

Higher density of defects at higher T

Defects represent disorder - disorder

favored at higher T Point defects

Vacant sites

Atoms out of correct position Extraneous atoms

Substituted atoms


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Crystal Defects

1. Point Defectsa) Schottky(vacancy) - seen with steel balls

in last frame

a) Need to maintain charge balance!

b) Impurity

Foreign ion replacesnormal one

Foreign ion is added (interstitial)

Both combined

a. Schottky defect

b. Interstitial (impurity) defect


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Line defects

Crystal deformation controlled by crystalstructure

Planes/locations are favored fordeformation based on bond strength

Bond breakage doesnt happen throughoutentire structure simultaneously

Lump in carpet


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Crystal Defects2. Line Defects

d) Edge dislocation

Migration aids ductile deformation

Fig 10-4 of

Bloss,

Crystallography

and Crystal

Chemistry.

MSA


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Crystal Defects

2. Line Defectse) Screw dislocation (aids mineral growth)

Fig 10-5 of

Bloss,

Crystallography

and Crystal

Chemistry.

MSA


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Planar defects

Mismatch of the crystal structure across a

surface

Officially grain boundaries count as planardefect


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Crystal Defects3. Plane Defects

f) Domainstructure (antiphasedomains)

Has short-range but not long-range order

Fig 10-2 of Bloss, Crystallography and Crystal Chemistry. MSA


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Crystal Defects

3. Plane Defects

g) Stacking faults

Common in clays and low-T disequilibrium

A - B - C layers may be various clay types (illite, smectite, etc.)

ABCABCABCABABCABC

AAAAAABAAAAAAA

ABABABABABCABABAB


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Line Defects

Dislocations

Missing half plane A Defect


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Missing half planeA Defect

An extra half plane


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An extra half plane

or a missing half plane


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What kind of

defect is this?

A line defect?

Or a planar defect?

An extra half plane


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An extra half plane

or a missing half plane

Edge

Dislocation


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This is a line defectcalled an

EDGE DISLOCATION


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1 2 3 4 5 6 7 8 9


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1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9


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1 2 3 4 5 6 7 8 9


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1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

slipno slip

boundary = edge dislocation

Slip planeb

Burgers vector


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Slip plane

slip no slip

dislocation

b

t

Dislocation: slip/no slip

boundary

b: Burgers vectormagnitude and

direction of the slip

t: unit vector tangent

to the dislocation line


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Dislocation Line:

A dislocation line is the boundary between slip and no

slip regions of a crystal

Burgers vector:

The magnitude and the direction of the slip is

represented by a vector bcalled the Burgers vector,

Line vector

A unit vector ttangent to the dislocation line is called a

tangent vector or the line vector.


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In general, there can be any angle between the Burgers

vector b(magnitude and the direction of slip) and the

line vector t (unit vector tangent to the dislocationline)

b t Edge dislocation

b t Screw dislocation

Mixed dislocationb t, b t


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b

t

b || t

1 23


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If b || t

Then parallel planes to the dislocation line lose their

distinct identity and become one continuous spiral ramp

Hence the name SCREW DISLOCATION

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