Atomic and Ionic Arrangements

Chapter 3a: Atomic and Ionic Arrangements

• This chapter addresses how atoms and ions assemble into solid structures.
• Discusses describing arrangements in a crystalline solid.
• Examines how material density and properties depend on structure.
• It also covers how crystal structures of ceramic materials differ from metals.

Materials and Packing

• First structural level above atomic structure involves the arrangement of atoms.
• Four degrees of order or periodicity can be defined in arrangements:
  • No order
  • Short Range Order (SRO)
  • Long-Range Order (LRO)
  • Liquid Crystals (LCs)

Degrees of Order

• No Order: Seen in monatomic gasses at high temperature and low pressure where atoms have no spatial relationship.
• Short Range Order (SRO): Atomic arrangement extends only to nearest neighbors, similar to liquids
  • Amorphous solids exhibit SRO and lack a defined structure
  • Typical examples include glasses, many plastics, and gel-like materials.

Long-Range Order (LRO)

• LRO has a repeating, periodic arrangement of atoms or molecules.
• Crystalline materials exhibit LRO:
  • Single crystal materials have LRO throughout, except at the surface.
  • Polycrystalline materials consist of many crystals (grains) with different orientations.

Liquid Crystals (LCs)

• Liquid crystals behave as amorphous, liquid-like materials in one state.
• LCs form a crystalline material when stimulated
  • Commonly found in LCD TVs

Crystal Structure Formation

• Time, temperature, and bonding play important roles in atomic arrangement in crystal structures.
• A molten unary metal solidifies below its melting temperature, dependent on bond strength.
• Solid form depends on bonding and cooling rate.
  • Fast cooling rates lead to amorphous structures.
  • Slow cooling rates enable single crystal formation.

Defining the Lattice

• Lattice: A collection of points known as lattice points
  • Points are arranged in a periodic pattern.
  • Can exist in 1D (1 lattice), 2D (5 lattices), and 3D (14 lattices).
• Basis/Motif: A group of atoms located in a particular way relative to each other and associated with the lattice points

Unit Cell

• Unit Cell: Basic repetitive unit of structure in a three-dimensional grid.
• Unit cell shapes in nature can be stacked to fill three-dimensional space.
• Unit cell is a lattice subdivision retaining the overall characteristics.
• Only specific shapes can be stacked to fill three-dimensional space.
• Applies similarly to polygons and tiling a plane

Bravais Lattices

• Bravais lattices include simple cubic, face-centered cubic, and body-centered cubic structures.
• Also includes simple tetragonal, body-centered tetragonal, and hexagonal structures.
• Other Bravais lattices: simple orthorhombic, body-centered orthorhombic, base-centered orthorhombic, face-centered orthorhombic, rhombohedral, simple monoclinic, base-centered monoclinic, and triclinic.

Lattice Parameters

• Lattice Parameters: Axial lengths/dimensions of the unit cell denoted by a, b, and c, measured in nanometers (nm) or angstroms (Å).
• Interaxial Angle: Angle between the axial length, denoted by α, β, and γ, following a strict convention for angles between lengths.

Equations for Cubic, Hexagonal, Tetragonal, Rhombohedral, and Orthorhombic Structures

• Cubic: a = b = c and α = β = γ = 90°, with Volumecubic ​​= a³.
• Hexagonal: a = b ≠ c and α = β = 90°, γ = 120°, with Volumehexagonal = 0.866a²⋅c.
• Tetragonal: a = b ≠ c and α = β = γ = 90°, with Volumetetragonal = a²⋅c.
• Rhombohedral: a = b = c and α = β = γ ≠ 90°, with Volumerhombohedral = a³√1 − 3 cos α² + 2 cos α³.
• Orthorhombic: a ≠ b ≠ c and α = β = γ = 90°, with Volumeorthorhombic = a⋅b⋅c.

Equations for Monoclinic, and Triclinic Structures

• Monoclinic: a ≠ b ≠ c, α = γ = 90° ≠ β, and Vmonoclinic = abc sin β.
• Triclinic: a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°, and Vtriclinic = abc√1 − cos α2 − cos β2 − cos γ2 + 2 cos α cos β cos γ.

Metallic Crystal Structures

• Tend to be densely packed because:
  • Typically only one element is present, all atom radii are the same.
  • Metallic bonding is non-directional
  • Nearest neighbor distances are small lowering bond energy
  • Electron cloud shields cores from each other.
• Metallic structures have the simplest crystal frameworks.

Energy and Packing

• Non-dense, random packing has higher energy state.
• Dense, ordered packing has lower energy state.
• Dense, ordered packed structures tend to have lower energies.

Atomic Hard Sphere Model

• Metallic crystal structures can be envisioned using the Atomic Hard Sphere Model.
• This model explores how to stack metal atoms to minimize empty space.

Simple Cubic Structure (SC)

• Closes-packed directions are cube edges.
• Coordination number is 6, meaning each atom has 6 nearest neighbors.
• One atom per unit cell (Atoms # /unit cell =1= 8 corners x 1/8).
• Rare due to low packing density; only Polonium (Po) has this structure.
• Atomic Packing Factor (PF) for a simple cubic structure = 0.52.

Body-Centered Cubic Structure (BCC)

• Atoms touch each other along cube diagonals.
• Coordination number is 8 (8 nearest neighbors).
• Two atoms per unit cell (Atoms # /unit cell = 2 = 1 center + 8 corners x 1/8).
• Examples: Chromium (Cr), Tungsten (W), Iron (Fe α), Tantalum, Molybdenum.
• Atomic Packing Factor for BCC is 0.68.

Face-Centered Cubic Structure (FCC)

• Atoms touch each other along face diagonals.
• Coordination # = 12 (12 nearest neighbors)
• Atoms # /unit cell = 4 = 6 face x 1/2 + 8 corners x 1/8.
• Examples: Aluminum (Al), Copper (Cu), Gold (Au), Lead (Pb), Nickel (Ni), Platinum (Pt), Silver (Ag).
• Atomic Packing Factor for FCC is 0.74.

Hexagonal Close-Packed (HCP)

• Has 3D and 2D projections with A and B sites
• Coordination # = 12
• Atoms # /unit cell = 2 (1 inside + 8 corners x 1/8)
• PF = 0.74 and c/a = 1.633
• FCC and HCP have the same PF (0.74) and coordination # (12)
• Examples: Cadmium (Cd), Magnesium (Mg), Titanium (Ti), Zinc (Zn)

Theoretical Density (ρ)

• Density = (Mass of Atoms in Unit Cell) / (Total Volume of Unit Cell).
• ρ = nA / VCNA, where n = # of atoms/unit cell, A = atomic mass, VC = Volume of unit cell (a³ for cubic), and NA = Avogadro's number (6.022 x 1023 atoms/mol).
• The calculation for material density is similar to packing factor, but atomic masses are used instead of atomic volumes.
• For Chromium (Cr) BCC: A = 52.00 g/mol, R = 0.125 nm, n = 2 atoms/unit cell, a = 4R/√3= 0.2887 nm.
• Theoretical density is 7.18 g/cm³, actual density is 7.19 g/cm³.

Densities of Material Classes

• In general: ρmetals > ρceramics > ρpolymers
• Metals have close-packing (metallic bonding) and often large atomic masses.
• Ceramics have less dense packing and often lighter elements.
• Polymers: Low packing density (often amorphous) and lighter elements (C, H, O).
• Composites have intermediate density values.

Allotropy

• Allotropy: Two or more distinct crystal structures for the same material (also known as polymorphism).
• For iron: BCC α-Fe exists at 912°C, FCC γ-Fe exists at 1394°C, and BCC δ-Fe exists at 1538°C (liquid).

Allotropic Transformations

• Allotropic Transformation: Change from one structure to another on heating and cooling; also known as polymorphic transformation
• Basis is important for control over iron and steel microstructure and properties.
• For Fe: BCC exists at room temperature while FCC exists at higher temperatures (above 912°C).
• Volume increases abruptly by more than 1% when cooling below 912°C and produces residual stresses.