Aula 3 - Crystalline Solids

The Structure of Crystalline Solids

Aula 3 - Crystalline Solids - 31 de marzo de 2014

Types of solids❖ Crystalline material: atoms self-

organize in a periodic array!

❖ Single crystal: atoms are in a periodic array over the entire extend of the material!

❖ Polycrystalline material: composed of small crystals or grains!

❖ Amorphous: disordered, lack of a systematic arrangement

�2


Crystal Structures

❖ Consider atoms to be hard-spheres with well defined radii. The shortest distance between two like atoms is one diameter of the hard sphere

The way in which atoms or molecules are arranged

2R

Crystalline Lattice: A tridimensional matrix

representing the position of the atoms

�3


Energy and Packing

�4

•Non dense, random packing

Chapter 3 - 2

• Non dense, random packing

• Dense, ordered packing

Dense, ordered packed structures tend to have lower energies.

Energy and Packing Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r



Chapter 3 - 2





r



Energy

r



•Dense, ordered packing

Chapter 3 - 2





r



Energy

r



Chapter 3 - 2





r



Energy

r



Dense, ordered packed structures, tend to have lover energies


Unit Cells

❖ The choice of origin of the repeating unit can vary!

❖ Most are represented by parallelepipeds or prisms

The smallest repeating unit which shows the full symmetry of the crystal structure

2-Dimensional

Na+ Cl+ Na+ Cl+

Na+Cl+ Na+Cl+

Na+ Cl+ Na+ Cl+

3-Dimensional

�5


Metallic Crystal Structures

�6

How can we stack atoms to minimize empty space?

Chapter 3 - 4

Metallic Crystal Structures •  How can we stack metal atoms to minimize

empty space? 2-dimensions

vs.

Now stack these 2-D layers to make 3-D structures

2 - dimensions

Stack these 2-D layers to make 3-D structures


Metallic Crystal Structures❖ Metals are usually polycrystalline!

❖ Compact arrangements!

❖ Using the hard sphere model, each sphere represents a ionic nucleus!

❖ Atomic (hard sphere) radius, R, defined by ion core radius - typically 0.1 - 0.2 nm

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fcc

hcp

bcc

face-center cubic

hexagonal close-packed

body-center cubic


Simple Cubic Structure (SC)

�8

Only Po has this estructure

•Rare due to low packing density!•Close-packed directions are cubic edges

Chapter 3 - 6

• Rare due to low packing density (only Po has this structure) • Close-packed directions are cube edges.

• Coordination # = 6 (# nearest neighbors)


Click once on image to start animation

(Courtesy P.M. Anderson)

Chapter 3 - 6

• Rare due to low packing density (only Po has this structure) • Close-packed directions are cube edges.

• Coordination # = 6 (# nearest neighbors)


Click once on image to start animation

(Courtesy P.M. Anderson)


Atomic Packing Factor (APF)

�9

APF = Sum of atomic volumes!Volume of cell

•APF for a simple cubic structure = 0,52

Chapter 3 - 7

• APF for a simple cubic structure = 0.52

APF = a 3

4 3 π (0.5a) 3 1

atoms unit cell

atom volume

unit cell volume


APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

Adapted from Fig. 3.3 (a), Callister & Rethwisch 9e.

close-packed directions

a

R = 0.5a

contains 8 x 1/8 = 1 atom/unit cell

Chapter 3 - 7

• APF for a simple cubic structure = 0.52

APF = a 3

4 3 π (0.5a) 3 1

atoms unit cell

atom volume

unit cell volume


APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

Adapted from Fig. 3.3 (a), Callister & Rethwisch 9e.

close-packed directions

a

R = 0.5a

contains 8 x 1/8 = 1 atom/unit cell

• APF: fraction of volume occupied by hard spheres


Body-Centered Cubic (bcc) Crystal Structures

❖ Cubic geometry!❖ Examples: Cr, Fe, W

• Atoms are located at each of the corners and on one another at the center of the cube

�10

a

•a: cube edge length!•R: atomic radius

cubic edge length!a = 4R /√3

Two atoms are associated with each BCC unit cell: the equivalent of one atomfrom the eight corners, each of which is shared among eight unit cells, and the sin-gle center atom, which is wholly contained within its cell. In addition, corner andcenter atom positions are equivalent. The coordination number for the BCC crys-tal structure is 8; each center atom has as nearest neighbors its eight corner atoms.Because the coordination number is less for BCC than FCC, so also is the atomicpacking factor for BCC lower—0.68 versus 0.74.

The Hexagonal Close-Packed Crystal StructureNot all metals have unit cells with cubic symmetry; the final common metallic crystalstructure to be discussed has a unit cell that is hexagonal. Figure 3.3a shows a reduced-sphere unit cell for this structure, which is termed hexagonal close-packed (HCP); an

3.4 Metallic Crystal Structures • 49

(a) (b) (c)

Figure 3.2 For the body-centered cubic crystal structure, (a) a hard-sphere unit cellrepresentation, (b) a reduced-sphere unit cell, and (c) an aggregate of many atoms. [Figure(c) from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties ofMaterials, Vol. I, Structure, p. 51. Copyright © 1964 by John Wiley & Sons, New York.Reprinted by permission of John Wiley & Sons, Inc.]

hexagonal close-packed (HCP)

c

aA

BC

J

E

G

H

F

D

(b)(a)

Figure 3.3 For the hexagonal close-packed crystal structure, (a) a reduced-sphere unitcell (a and c represent the short and long edge lengths, respectively), and (b) an aggregateof many atoms. [Figure (b) from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structureand Properties of Materials, Vol. I, Structure, p. 51. Copyright © 1964 by John Wiley & Sons,New York. Reprinted by permission of John Wiley & Sons, Inc.]

Crystal Systems andUnit Cells for Metals

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JWCL187_ch03_044-089.qxd 11/9/09 9:31 AM Page 49




(a) (b) (c)



c

aA

BC

J

E

G

H

F

D

(b)(a)



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Body-Centered Cubic (bcc) Crystal Structures


�11

11MSE 2090: Introduction to Materials Science Chapter 3, Structure of solids

Body-Centered Cubic (BCC) Crystal Structure (I)

Atom at each corner and at center of cubic unit cellCr, D-Fe, Mo have this crystal structure

R

Atoms /Unit cell = 2

Coordination Number: 8

a


BCC - APF

�12Chapter 3 - 10

Atomic Packing Factor: BCC

APF =

4 3 π ( 3 a/4 ) 3 2

atoms unit cell atom

volume

a 3 unit cell volume

length = 4R = Close-packed directions:

3 a

• APF for a body-centered cubic structure = 0.68

a R Adapted from Fig. 3.2(a), Callister & Rethwisch 9e.

a

a 2

a 3 APF = 0,68


Face-Centered Cubic (fcc) Crystal Structures

�13

•All atoms in the unit cell are equivalent!

•FCC can be represented by a stack of close-packed planes (planes with highest density of atoms)


Face-Centered Cubic (FCC) Crystal Structure (I)

¾ Atoms are located at each of the corners and on the centers of all the faces of cubic unit cell

¾ Cu, Al, Ag, Au have this crystal structure

Two representations of the FCC unit cell R

a



❖ Cubic geometry!❖ Examples: Cu, Al, Ag, Au

• Atoms are located at each of the corners and on the of all the faces of cubic unit cell

�14

a

•a: cube edge length!•R: atomic radius

cubic edge length!a = 2R√2

46 • Chapter 3 / The Structure of Crystalline Solids46 • Chapter 3 / The Structure of Crystalline Solids

(a) (b)

(c)

Figure 3.1 For the face-centered cubic crystalstructure, (a) a hard-sphere unit cellrepresentation, (b) areduced-sphere unit cell, and(c) an aggregate of manyatoms. [Figure (c) adaptedfrom W. G. Moffatt, G. W.Pearsall, and J. Wulff, TheStructure and Properties ofMaterials, Vol. I, Structure, p.51. Copyright © 1964 byJohn Wiley & Sons, NewYork. Reprinted bypermission of John Wiley &Sons, Inc.]

Cr yst a l St ructures3.2 FUNDAMENTAL CONCEPTS

Solid materials may be classified according to the regularity with which atoms or ionsare arranged with respect to one another. A crystalline material is one in which theatoms are situated in a repeating or periodic array over large atomic distances; thatis, long-range order exists, such that upon solidification, the atoms will position them-selves in a repetitive three-dimensional pattern, in which each atom is bonded to itsnearest-neighbor atoms. All metals, many ceramic materials, and certain polymersform crystalline structures under normal solidification conditions. For those that donot crystallize, this long-range atomic order is absent; these noncrystalline or amor-phous materials are discussed briefly at the end of this chapter.

Some of the properties of crystalline solids depend on the crystal structure ofthe material, the manner in which atoms, ions, or molecules are spatially arranged.There is an extremely large number of different crystal structures all having long-rangeatomic order; these vary from relatively simple structures for metals to exceedinglycomplex ones, as displayed by some of the ceramic and polymeric materials. The pres-ent discussion deals with several common metallic crystal structures. Chapters 12 and14 are devoted to crystal structures for ceramics and polymers, respectively.

When describing crystalline structures, atoms (or ions) are thought of as beingsolid spheres having well-defined diameters. This is termed the atomic hard-sphere

crystalline

crystal structure



(a) (b)

(c)






crystalline

crystal structure




❖ Cubic geometry!❖ Examples: Cu, Al, Ag, Au, Pb,

Ni, Pt

�15

CN: 12

• Coordination number (CN): the number of closest neighbors to which

an atom is bonded



(a) (b)

(c)






crystalline

crystal structure


a


FCC - APF

�16Chapter 3 - 12

• APF for a face-centered cubic structure = 0.74 Atomic Packing Factor: FCC

maximum achievable APF

APF =

4 3 π ( 2 a/4 ) 3 4

atoms unit cell atom

volume

a 3 unit cell volume

Close-packed directions: length = 4R = 2 a

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a

2 a

Adapted from Fig. 3.1(a), Callister & Rethwisch 9e.

APF= 0,74


Closed-Packed Structures

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A A A A A A

A A A A A A

A A A A A A

A

A A A A A

B B B B B B

B B B B B B

B B B B B B

B

B B B B B

C C C C C C

C C C C C C

C C C C C C

C

C C C C C

fcc: ABCABC... stacking sequence


Closed-Packed Structures

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A A A A A A

A A A A A A

A A A A A A

A

A A A A A

B B B B B B

B B B B B B

B B B B B B

B

B B B B B

A A A A A A

A A A A A A

A A A A A A

A

A A A A A

stacking sequence ABAB… structure hcp


Hexagonal Closed-Packed (hcp) Crystal Structures

❖ Hexagonal geometry!❖ Composed with various hc

unit cells!❖ Examples: Cd, Mg, Zn, Ti

•Six atoms form regular hexagon, surrounding one atom in center. !!

•Another plane is situated halfway up unit cell (c-axis), with 3 additional atoms situated at interstices of hexagonal (close-packed) planes

�19


Hexagonal Close-Packed Crystal Structure (I)

¾ HCP is one more common structure of metallic crystals

¾ Six atoms form regular hexagon, surrounding one atom in center. Another plane is situated halfway up unit cell (c-axis), with 3 additional atoms situated at interstices of hexagonal (close-packed) planes

¾ Cd, Mg, Zn, Ti have this crystal structure

a

c

•Two cell lattice parameters: a and c




(a) (b) (c)



c

aA

BC

J

E

G

H

F

D

(b)(a)



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Hexagonal Closed-Packed (hcp) Crystal Structures


�20


Coordination Number: 12

APF: 0,74











a

c

Ideal c/a = 1,633


Theoretical Density

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ρ = nA ! VCNAWhere:

n ! = !! number of atoms in each number cell!A != ! ! atomic weight!VC!= ! ! volume of the unit cell (a3 for cubic)!NA!= ! ! Avogadro’s number (6,022 x 1023 atoms/mol)


Theoretical Density, ρ

�22Chapter 3 - 17

•  Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n = 2 atoms/unit cell

ρtheoretical

a = 4R/ 3 = 0.2887 nm

ρactual

a R

ρ = a 3

52.00 2 atoms

unit cell mol g

unit cell volume atoms

mol

6.022 x 1023

Theoretical Density, ρ

= 7.18 g/cm3

= 7.19 g/cm3

Adapted from Fig. 3.2(a), Callister & Rethwisch 9e.


Densities of Different Materials

�23Chapter 3 - 18

Densities of Material Classes ρ metals > ρ ceramics > ρ polymers

Why?

Data from Table B.1, Callister & Rethwisch, 9e.

ρ (g

/cm

)

3

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers Polymers

1

2

2 0 30

B ased on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass,

Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers

in an epoxy matrix). 10

3 4 5

0.3 0.4 0.5

Magnesium Aluminum

Steels

Titanium

Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum

G raphite Silicon Glass - soda Concrete Si nitride Diamond Al oxide Zirconia

H DPE, PS PP, LDPE PC

PTFE

PET PVC Silicone

Wood

AFRE * CFRE * GFRE* Glass fibers Carbon fibers A ramid fibers

Metals have... • close-packing (metallic bonding) • often large atomic masses Ceramics have... • less dense packing • often lighter elements Polymers have... • low packing density (often amorphous) • lighter elements (C,H,O) Composites have... • intermediate values

In general


Single Crystals❖ Atoms are in a repeating or periodic array over the

entire extent of the material

�24

•All unit cells present the same orientation and are linked in the same way to each other!

•Natural or artificially grown!

•Has a geometric regular shape


Polycrystalline material❖ Comprised of many small crystals or grains. !

❖ Grain sizes between 1nm - 2 cm!

❖ The grains have different crystallographic orientation. !

❖ Overall component properties are not directional!

❖ There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries.

�25


Materials Solidification

�26


Single Crystals and Polycrystalline Materials

Single crystal: atoms are in a repeating or periodic array over the entire extent of the material

Polycrystalline material: comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries.

Grain Boundary


Single Crystals vs Polycrystals

�27

Single Crystals•Properties vary with direction: anisotropic!

•Example: the modulus of elasticity in BCC iron

Chapter 3 - 21

• Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron:

Data from Table 3.4, Callister & Rethwisch 9e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

• Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic.

200 µm Adapted from Fig. 4.15(b), Callister & Rethwisch 9e. [Fig. 4.15(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC (now the National Institute of Standards and Technology, Gaithersburg, MD).]

Single vs Polycrystals E (diagonal) = 273 GPa

E (edge) = 125 GPa Polycrystals

•Properties may/may not vary with direction!

•If grains are randomly oriented: isotropic!

•If grains are textured: anisotropic

Chapter 3 - 21

• Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron:

Data from Table 3.4, Callister & Rethwisch 9e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

• Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic.

200 µm Adapted from Fig. 4.15(b), Callister & Rethwisch 9e. [Fig. 4.15(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC (now the National Institute of Standards and Technology, Gaithersburg, MD).]

Single vs Polycrystals E (diagonal) = 273 GPa

E (edge) = 125 GPa


Polymorphism

❖ Can be found in any crystalline materials, including metals, minerals, polymers!

❖ Relevant to the fields of pharmaceuticals, agrochemicals, pigments, dyestuffs, foods, and explosives

�28

• Polymorphism: When a material exist in more than one crystal structure

quartz

cristobalite


Allotropy

�29

•Allotropy: Polymorphisme in an elemental solid

In Carbon

Diamond Graphite Fullerenes Carbon Black


Crystal Systems

�30

•Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal

Crystals Systems•Classification of crystal structures based on the geometry of the

unit cell, independent of the atomic positions in the cell

Substitution for the various parameters into Equation 3.5 yields

The literature value for the density of copper is 8.94 g/cm3, which is in veryclose agreement with the foregoing result.

3.6 POLYMORPHISM AND ALLOTROPYSome metals, as well as nonmetals, may have more than one crystal structure, a phe-nomenon known as polymorphism. When found in elemental solids, the condition isoften termed allotropy. The prevailing crystal structure depends on both the temper-ature and the external pressure. One familiar example is found in carbon: graphite isthe stable polymorph at ambient conditions, whereas diamond is formed at extremelyhigh pressures.Also, pure iron has a BCC crystal structure at room temperature, whichchanges to FCC iron at 912!C (1674!F). Most often a modification of the density andother physical properties accompanies a polymorphic transformation.

3.7 CRYSTAL SYSTEMSBecause there are many different possible crystal structures, it is sometimes conven-ient to divide them into groups according to unit cell configurations and/or atomicarrangements. One such scheme is based on the unit cell geometry, that is, the shapeof the appropriate unit cell parallelepiped without regard to the atomic positions inthe cell. Within this framework, an xyz coordinate system is established with its originat one of the unit cell corners; each of the x, y, and z axes coincides with one of thethree parallelepiped edges that extend from this corner, as illustrated in Figure 3.4.The unit cell geometry is completely defined in terms of six parameters: the three edgelengths a, b, and c, and the three interaxial angles a, b, and g. These are indicated inFigure 3.4, and are sometimes termed the lattice parameters of a crystal structure.

On this basis there are seven different possible combinations of a, b, and c, anda, b, and g, each of which represents a distinct crystal system. These seven crystalsystems are cubic, tetragonal, hexagonal, orthorhombic, rhombohedral,2 monoclinic,and triclinic. The lattice parameter relationships and unit cell sketches for each are

" 8.89 g/cm3

"14 atoms/unit cell 2 163.5 g/mol 2

[161211.28 # 10$8 cm 2 3/unit cell ] 16.022 # 1023 atoms/mol 2r "

nACu

VCNA"

nACu116R3 12 2NA


polymorphism

allotropy

2Also called trigonal.

z

y

x

a

!

b

c"

#

Figure 3.4 A unit cell with x, y, and z coordinate axes,showing axial lengths (a, b, and c) and interaxial angles (!, #, and ").

lattice parameters

crystal system


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•Unit cell geometry defined by:!•The three edge lengths: a, b, c!•The three interaxial angles: α, β, γ

Lattice Parameters

7 crystal systems


Crystal Systems54 • Chapter 3 / The Structure of Crystalline Solids

Table 3.2 Lattice Parameter Relationships and Figures Showing Unit CellGeometries for the Seven Crystal Systems

AxialCrystal System Relationships Interaxial Angles Unit Cell Geometry

Cubic

Hexagonal

Tetragonal

Rhombohedral (Trigonal)

Orthorhombic

Monoclinic

Triclinic a ! b ! g ! 90°a ! b ! c

a " g " 90° ! ba ! b ! c

a " b " g " 90°a ! b ! c

a " b " g ! 90°a " b " c

a " b " g " 90°a " b ! c

a " b " 90°, g " 120°a " b ! c

a " b " g " 90°a " b " c

ab

c!

"

#

ab

c!

ab

c

aaa

"

aa

c

aaa

c

aa

a


54 • Chapter 3 / The Structure of Crystalline Solids



Cubic

Hexagonal

Tetragonal


Orthorhombic

Monoclinic


a " g " 90° ! ba ! b ! c

a " b " g " 90°a ! b ! c

a " b " g ! 90°a " b " c

a " b " g " 90°a " b ! c

a " b " 90°, g " 120°a " b ! c

a " b " g " 90°a " b " c

ab

c!

"

#

ab

c!

ab

c

aaa

"

aa

c

aaa

c

aa

a


54 • Chapter 3 / The Structure of Crystalline Solids



Cubic

Hexagonal

Tetragonal


Orthorhombic

Monoclinic


a " g " 90° ! ba ! b ! c

a " b " g " 90°a ! b ! c

a " b " g ! 90°a " b " c

a " b " g " 90°a " b ! c

a " b " 90°, g " 120°a " b ! c

a " b " g " 90°a " b " c

ab

c!

"

#

ab

c!

ab

c

aaa

"

aa

c

aaa

c

aa

a



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Aula 3 - Crystalline Solids