Elements of Materials Science and Engineering ChE 210 Chapter 3

Crystalline Phases Phase: is that part of a material that is recognized from other parts in structure and composition, e.g. ice-water The same composition but different in state Phase boundary: between the two states locates the discontinuinity in structure Salt and saltwater (brine), both contain NaCl but form different phases (solid/solution) Alcohol-water (miscible) different composition but form one phase Oil-water (immiscible): two phases

Crystals Metals, ceramics and certain polymers crystallize by solidification Crystalline phases: Posses a periodicity of long range order i.e. The local atomic arrangement is repeated at regular intervals millions of times in 3-D of space.

Crystalline structure (NaCl) The atomic (or ionic) coordination is repeated to give a long range periodicity. The centers of all the cube faces duplicate the pattern at the cube corners. At the center of each cell there is a Na+

Crystal Systems Three dimensional system: x, y, z axes and , , and  are the axial angles A, b, and c are the unit cell dimensions Crystal systems vary in the axial angles and the unit cell dimensions

Crystal Systems  =  = 90O ≠  Axial Angle Axes System a = b = c Cubic a = b ≠ c Tetragonal a ≠ b ≠ c Monoclinic  =  = 90O ≠  Triclinic  ≠  ≠  ≠ 90O orthorhombic  =  = 90O;  =120O a = a ≠ c Hexagonal  =  =  ≠ 90O rhombhedral

Noncubic unit cell a a b c a=b≠c, angles =90o Tetragonal system a = a ≠ c, angles =90o, 120o hexagonal system a ≠ b≠c, angles =90o orthorhombic system

Lattices The above 7 systems can be divided into 14 patterns of points, called Bravais lattices. e.g Cubic systems: Body centered cubic (BCC) Face centered cubic (FCC) Simple cubic

FCC lattice, e.g. solid methane. CH4 solidifies @ -183oC between 20-90 K molecules rotate in their lattice sites. Below 20 K, the molecules have identical alignments, as shown here.

Example 3-1.1: the unit cell of Cr metal is cubic and contains 2 atoms, if the density of Cr = 7.19 Mg/m3; determine the dimension of Cr unit cell Mass of a unit cell = 2 Cr(52.0 g)/(6.02 x 1023 Cr) = 172.76 x 10-24 g volume = a3 = (172.76 x 10-24 g)(7.19 x 106 g/m3) = 24 x 10-30 m3 a = 0.2884 x 10-9 m

Cubic System (Body-centered cubic metals) BCC of a metal Show the location of the atom centers. Model made from hard balls

(b.d) bcc metal = 4R = abcc metal √3 Body centered (bcc) Metal structure: two metal atoms/unit cell and the atomic packing factor of 0.68. Since atoms are in contact along the body diagonals (b.d.) of the unit cell: (b.d) bcc metal = 4R = abcc metal √3 abcc metal = 4R/√3 a is the lattice constant, R = the radius of the ion

The atomic packing factor (PF) of a bcc metal assuming spherical atoms (hard ball model) and calculating the volume fraction of the unit cell that is occupied by the atoms There are 2 atoms per unit cell in a bcc metal, and we assuming spherical atoms

Face-centered cubic metals Fcc metal structure, a) Schematic view showing location of atom centers, b) hard ball model. Each unit cell contains 4 atoms (f.d.) fcc metal = 4R = afcc metal √2 afcc metal = 4R/√2 and the PF = 0.74 This is because each atom has 12 neighbours

FCC metal structure: 4 metal atoms/unit cell atomic packing = 0.74 atoms are in contact along the face diagonals i.e. afcc = 4R/√2, This is not applied to all fcc systems

Other face-centered cubic structure e.g. molecular structure of solid methane with 5 atoms occupying the fcc positions as a molecular unit. NaCl structure, since the atoms are unlike. afcc NaCl = 2(rNa+ + RCl-) Note that there is a gap between the Cl- ions, we can not apply the eqn used for lattice constant.

Diamond (fcc)

Simple cubic structures It can be more complex than fcc or bcc Simple means a crystal structure with repetition at only full unit cell increments. bcc structure is replicated at the centre of the unit cell as well as the corners. fcc is replicated at the face centers as well as the corners. A simple cubic (sc) cell lacks at both these centered locations. In CsCl, (sc), Cl- at the cell corners; Cs+ at the cell centre. But this is not bcc , because the cell centre possesses an ion different from that at the cell corner.

Simple cubic structures

Properties of simple cubic structures Binary (Two component system) The unit cell has two radii. aCsCl-type = 2(rCs+ + RCl-)/√3 CN = 8, r/R = 0.73, not in case of MgO rMg2+/RO2- = 0.47 Calcium titanate, CaTiO3 is also SC - Ca2+ are located at the corners of the unit cell Ti4+ are located at the center O2- at the center of each face Therefore, it is neither bcc nor fcc

Example 3-21 Calculate (a) the atomic packing factor of an fcc metal (b) the ionic packing factor of fcc NaCl There are 4 atoms/unit cell in an fcc metal structure - But there are 4 Na+ and 4 Cl- per unit cell. - Note: The lattice constants is related differently from fcc

Example 3-2.2 p71 Copper has an fcc metal structure with an atom radius of 0.1278 nm. Calculate its density and check this value with the density listed in Appendix B. Procedure: Since  = m/V, we must obtain the mass per unit cell and the volume of each unit cell. The former is calculated from the mass of four atoms; the latter requires the calculation of the lattice constant, a, for an fcc metal structure.

Example 3-2.2 p71

Hexagonal Close-Packed Metals Noncubic structures Hexagonal Close-Packed Metals

Noncubic structures Hexagonal Close-Packed Metals Examples are Mg, Ti and Zn Called the hexagonal closed-packed structure (hcp) Each atom is located directly above or below interstices among three atoms in the adjacent layer below its plane Each atom touches 3 atoms below and 6 atoms in the same plane. Average of 6 atoms/ unit cell. Such as the fcc CN =12 Therefore, the packing factor = 0.74

Other non cubic structure Iodine  orthorhombic, because I2 is not spherically sym. Polyethylene  orthorhombic, complex, due to the large and linear molecules. Graphite  hexagonal structure

Iron carbide: Called cementite., Fe3C The crystal lattice contains 2 elements in a fixed 3:1 ratio Orthorhombic, 12 Fe and 4 C atoms in each unit cell. Carbon in the interstices among the larger Fe atoms. it greatly affects the properties of steel due to its superior hardness.

Barium Titanate: BaTiO3, Tetragonal system Tetragonal BaTiO3: The base of the unit cell is square; but unlike in CaTiO3, c ≠ a. Also, the Ti4+ and O2- ions are not symmetrically located with respect to the corner Ba2+ ions. Above 120oC, BaTiO3 changes from this structure to that of CaTiO3.

Fig. 3-3.2

Polymorphism Diamond Graphite c The existence of two crystal structure of the same substance is called polymorphism. The two forms are called polymorphs. Heat + pressure Diamond Graphite c

bcc fcc CN of bcc Fe = 8 at room temp., PF = 0.68 and R = 0.1241 nm CN of fcc = 12 at 912oC, PF = 0.74 and R = 0.129 nm For bcc iron R = 0.126 due to thermal expansion The reaction involves volume changes and 1.4 V% (-0.47 l%) when the temperature is raised above 912oC. Volume changes is due to larger CN in fcc atoms

Zirconia toughen alumina significantly, because it has 3 polymorphs At high temperature changes from cubic to tetragonal with CaF2 structure The next step it changes to monoclinic, involves dimensional strain that consumes energy before fracture

Unit Cell Geometry X- axis points to us. Y-axis points to our right Z- axis points up ward Origin is the lower left corner Every point is identified by coefficients, Thus the origin is 0,0,0. The centre of the unit cell is at a/2, b/2, c/2 The indices of that location are ½, ½, ½ The far corner of the unit cell is 1,1,1, regardless of the crystal system.

Crystal indices Transition from a selected site within the unit cell by integer multiple of lattice constants. Replication occurs by ±a/2, ±b/2, ± c/2 in body centered systems. i.e. a point with indices 0.3, 0.4, 0.2 duplicates at 0.8, 0.9, 0.7 and 0.8, -0.1, 0.7. In face centered systems, replication occurs by translation of ±a/2, ±b/2, 0 of ±a/2, 0, ±c/2 and of 0, ±b/2, ±c/2.

Crystal directions Can be represented by a vector from the origin to a point with the lowest integer index, h,k,l. Specified by indices [h,k,l]

All the parallel directions use the same label or index The parallel line that passes through 0,0,0 and since there are millions of points on any line. We choose the point with lowest set of integers. [111] direction passes from 0,0,0 through 1,1,1 ans ½, ½, ½. Likewise, the [112] passes through ½,1/2,1/2. Direction indices are written in square brackets [uvw], no commas, it can be positive or negative. Negative directions are indicated by over bar.

Crystal directions

Angles between directions The angle between [110] and [112] i.e. [110]  [112]) is the In cubic crystals, we can determine cos [uvw]  [u`v`w`] by dot product Repetition Distances The vector between identical positions within a crystal It differs from direction to direction and from structure to structure Example: in bcc metal, there is a repetition each 2R (which is a√3/2). The repetition distance is a√2 in the [110] direction of bcc but a/√2 in fcc.

Families of Directions In the cubic system: Are the same

Crystal planes Is defined by miller indices (hkl) It is the reciprocals of the intersections plane axes. Take the smallest possible integers Negative is indicated by over bar. Visualized as those outline the unit cell All parallel planes are identified by the same indices.

Crystal planes

X-ray diffraction X-rays is electromagnetic radiation but with very short wavelength compared to visible light. Monochromatic x-rays have one and the same wavelength. White x-rays have different wavelengths. When two x-rays beams lie in plane a constructive interference occurs. i.e. P.d. = n 

Bragg’s Law When monochromatic x-ray diffracted from two crystallographic planes n .  = 2 . D. sin