Lecture 21 MNS 102: Techniques for Materials and Nano Sciences

• Module 2: Basic Metrology and Materials Characterization: for structures, composition, and other properties (thermal, electrical, magnetic).

• Structures: Packing, grains, polymorphism.

• Basic concepts for crystal structure: point group symmetry; symmetry classes and Bravais lattices.

• Miller indices, crystallographic planes.

• Structures of metals, ceramics, polymers.

• Densities of Materials

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Reference: #4 W. D. Callister, "Materials Science and Engineering: An Introduction", 7th ed., Wiley, New York (2006), Ch. 3, 12, 14. - Also other editions. Also: http://www.learncheme.com/page/callistered4ch3

Homework 3B: Read the reference: W. D. Callister, "Materials Science and Engineering: An Introduction", 7th ed., Wiley, New York (2006), Ch. 3, 12, 14, or equivalent chapters on the structures of crystalline solids (metals), ceramics and polymers in other editions or textbooks.

Packing to Minimize Energy

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Energy

r

typical neighbour bond length

typical neighbour bond energy

Energy

r

typical neighbour bond length

typical neighbour bond energy

• Packing is a thermodynamically driven process.

• Random packing – higher in energy

• Ordered packing – energy minimum

• Dense, ordered packed structures tend to have lower energies.

• Crystalline materials: atoms (or groups of atoms) are arranged in a periodic array over large atomic distances. E.g. All metals, ceramics, some polymers.

• Noncrystalline or amorphous materials: no long-range ordering.

Crystalline SiO2

Noncrystalline SiO2

Grains and Polycrystals

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• Most materials are polycrystalline, consisting of grains.

• A grain or crystallite is a single crystal.

• Grains are randomly oriented, making the overall properties not directional

• Grain size ranges from 1 nm to cm

• Defects are important.

Source: http://mmmpdl.snu.ac.kr/?p=374

Allotropy/Polymorphism and Phase Diagrams

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Two or more distinct crystal structures for the same materials.

e.g.

Diamond vs graphite

Crystal Structure

• Crystal structure = Periodic arrangement of atoms or molecules or ions in 3D.

• Lattice = 3D array of points for individual atom positions.

• Unit cell = Repeating parallelepiped unit (or building block) used to generate the lattice, and defined by axial lengths (a, b, c) and interaxial angles (α, β, γ)

• Point group symmetry: For point group, there are five symmetry operations corresponding to symmetry elements (point, line, or plane). “Point” group means group with at least one point invariant in space after operation. “Group” comes from mathematics – members of the group follow certain rules (closure, associativity, identity, inverse, multiplication) and be able to transform from one to another.

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Homework 3C: Review the following websites and learn about point group symmetry. How would the symmetry elements change from benzene to chlorobenzene? http://www.facstaff.bucknell.edu/kastner/xray2008/CountSymmetryElements/PointGroupSymmetry/PointGroupSymmetry.html http://symmetry.otterbein.edu/tutorial/pointgroups.html

Point Group Symmetry

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Element Operation Symbol Examples

Any axis Identity – Do nothing E or C1 CHFClBr, SOClBr

Symmetry plane

Mirror reflection through plane NH3, H2O

Inversion center

Inversion through a center point i.e. (x, y, z) → (-x, -y, -z)

i

Proper axis Proper rotation by 360/n Cn NH3 C3

Improper axis

Improper rotation by (a) rotation by 360/n and (b) mirror reflection through the plane perpendicular to rotation axis

Sn CH4 S4

Seven (7) Symmetry Classes

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In group theory…

Symmetry class = a complete set of symmetry elements that are conjugate with each other. i.e.

If A, B, and X are group elements, then A and B are conjugates if A = X-1 B X and B = X-1 A X

(similarity transform).

More in upper year Math courses…

Fourteen (14) Bravais Lattices

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A Bravais lattice is a distinct lattice type defined by a set of relations among the six lattice parameters.

P=primitive I = body-centered F = face-centered C = side-centered

http://www.youtube.com/watch?v=NYVSI83KiKU

Miller Indices

A vector r from the origin to a lattice point can be written as: r = x a + y b + z c where a, b, c are unit vectors.

Miller indices are used to specify directions and planes in lattices or in crystals.

These planes are important b/c they affect optical properties, reactivity, surface tension, dislocation, and other surface-related properties.

Notation

• (h, k, l) represents a point – note exclusive use of commas for a point (negative numbers denote negative directions and written with a bar on top of the number)

• [h k l] represents a direction

• <h k l> represents a family of directions

• (h k l) represents a plane

• {h k l} represents a family of planes

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Recipe for Miller Indices

1. Identify the plane intercepts on the x, y, z axes.

2. Rewrite the intercepts in fractional coordinates.

3. Clear the fraction by taking the reciprocals of the fractional intercepts.

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Miller Indices for Hexagonal Class

Homework 3D: Watch the following screencasts:

• http://www.youtube.com/watch?v=pMTA_wiY784&feature=youtu.be

• http://www.youtube.com/watch?v=vK913oWl_XI&feature=youtu.be

Determine the Miller indices for the planes shown below [both shaded planes (A and B) for the cubic unit cell and the shaded plane for the hexagonal unit cell] and calculate their interplanar spacings.

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Crystallographic Planes

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Structures of Metals

• Metals ~ ion cores in a sea of electrons.

• Metallic bonding is nondirectional > no restriction for the number and position of the nearest neighbour atoms.

• Three common types: body-centered cubic (BCC); face-centered cubic (FCC); hexagonal close-packed (HCP). [FCC is also called CCP (cubic close-packed)].

• Simple cubic? Uncommon, and not stable packing. Just Polonium (Po).

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NB Early lanthanides adopt HC (4H). HC (4H) type has a complicated packing sequence such as ABAC, ABCB, etc. Two types of atom arrangement: H – anticuboctahedral like HCP; C – cuboctahedral CCP; plus 4 Hexagonal layers in c-direction > 4H.

NB Sm (samarium) has an even more complicated sequence of ACACBCBAB ACACBCBAB .... 21- 14

Cubic Class

• Counting atoms for the cubic unit cell:

Vertex atom shared by 8 cells > 1/8 atom per cell.

Edge atom shared by 4 cells > 1/4 atom per cell.

Face atom shared by 2 cells > 1/2 atom per cell.

• Coordination Number (CN) = number of nearest neighbours

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Atomic Packing Fraction (APF)

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FCC or CCP

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HCP

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CN = 12 6 atoms per unit cell a = 2 R c = 1.633 a (ideal case) APF = 0.74

Homework 3E: Download the handout on HCP. Work through and understand the derivation. Now, using the Rhenium (Re) result for VC obtained in the last part of the handout, work out the actual APF. Note that for Re, c = 1.615 a.

CCP vs HCP

Repeating along <111> direction

CCP or FCC: ABC ABC ABC…

HCP: AB AB AB…

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Linear Density, Planar Density, Theoretical Density

• Linear density = No. of atoms along a direction vector per length of direction vector

Linear packing density = No. of radii along a direction vector per length of direction vector

e.g. LD along [110] = 2 atoms / 4R = 1/(2R) LPD along [110] = 2 (2R) / 4R = 1

LD along [100] = 2/(4R) – Do as an exercise.

• Planar density = No. of atoms per area of plane Planar packing density = No. of atomic areas per area of plane e.g. PD on the (110) plane = 2 atoms / (82R2) = 1/ (42 R2) PPD on the (110) plane = 2 ( R2) / (82R2) = / (42)

• Theoretical density = Mass per volume = (No. of atoms per unit cell x Weight of atom)/(Volume of unit cell)

where weight of atom = atomic weight / Avogadro’s number

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Densities of Materials

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Non-metallic FCC and BCC Materials

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Structures of Ceramics

• Ceramics structures are more complex than those of metals b/c they are composed of more than one elements.

• Bonding may range from purely ionic (non-directional) to totally covalent (directional).

• % ionic character = {1 exp[(0.25)(XA XB)2]} 100 where XA, XB are electronegativities of element A and B.

• For total ionic bonding: structure depends on (a) magnitude of electrical charge on each of the ions and (b) the relative sizes of the cations (+’ve ions) and anions (-’ve ions), plus structure must be electrically neutral.

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Electronegativity Scale

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• Cations are usually smaller than anions, i.e. rC < rA or rC /rA < 1.

• Cations and anions prefer to have as many neighbouring ions as possible.

• Stable structure is obtained when anions (surrounding a cation) are all in contact with that cation.

• CN (Coordination Number) depends on rC /rA and there is a critical minimum rC /rA.

• Theoretical density = Mass per volume = (No. of FU per unit cell x Weight of FU)/(Volume of unit cell)

where FU = formula unit, and

weight of FU = ( atomic weights of all cations + atomic weights of all anions) / Avogadro’s number

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Common Ceramics

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NaCl, CsCl, Zinc Blende, Fluoride

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Perovskite & Spinel

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Complex Oxides

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Polymers

• Molecular entanglements occur when chains assume twisted, coiled and kinked shapes.

• Linear, branched, crosslinked and network structures possible. Copolymers include random, alternating, block and graft types.

• Amorphous in general, but could also exhibit varying degree of crystallinity – the latter usually interdispersed within amorphous regions. Polymers are chemically simple and have regular and symmetrical chain structures in order for crystallinity to occur.

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Polymer Crystals

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Common Polymers

Densities of Materials

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

Data from Table B1, Callister 6e.

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