Materials Notes PDF

The Structure of Crystalline Solids (How toms arrange in space) Energy and Packing - Non dense random packing vs dense ordered packing - Dense ordered materials tend to have lower energies - System wants lowest energy achievable so given enough time element will rearrange to dense ord...

The Structure of Crystalline Solids (How toms arrange in space) Energy and Packing - Non dense random packing vs dense ordered packing - Dense ordered materials tend to have lower energies - System wants lowest energy achievable so given enough time element will rearrange to dense order packing to achieve a lower energy Crystalline Materials - Atoms that are packed in periodic, 3D arrays - This is typical for metals, many ceramics and some polymers - Can be repeated over and over due to order and pattern Noncrystalline Materials - Atoms have no periodic packing so cant be repeated - Amorphous material = non crystalline material - Occurs for complex structures and rapid cooling of materials - Cooling of materials rapidly means that the atoms don’t have time to rearrange properly eg can’t reach equilibrium in time - This in turn will change the properties  - Avoid working with this in civil engineering due to its unpredictability and fragileness Unit Cell - A unit cell is a block of atoms which repeats itself to form space lattice - Materials arranged in short range order are called amorphous materials - The four basic types of unit cells are: - Simple - Body Centered - Face Centered - Base Centered Crystal Systems and Bravais Lattice - There are only seven different types of unit cells that are necessary to create all point lattices. - According to Bravais, (1811 to 1863) there are fourteen standard unit cells arrangements that can describe all possible lattice networks - Seven crystal systems:  - Bravais Lattices’s arrangements: - Simple vs face centered vs body centered unit cell: Principal Metallic Crystal Structures  - 90% of the metals have either body centered cubic (BCC), face centered cubic (FCC) or Hexagonal Close Packed (HCP) crystal structures - HCP is a denser version of simple hexagonal crystal structure - Examples: - Iron and chromium crystallizes in BCC structure - Aluminum, gold, silver, cooper crystallize in FCC structure - Titantim, magnesium, zinc crystallize in HCP structure Body Centered Cubic (BCC) Crystal Structure - Represented as one atom of each corner of cube and one at the center of the cube - Each atom has 8 nearest neighbors, so therefore the combination number is 8 - Corner atoms contact 8 other unit cells when repeated - Each unit cell has 1 of 8 atom at corners, therefore each unit cell has: Where 1 is atom at center, 8 represent 8 corners and ⅛ of each atom at corner - Atoms contact each other at cube diagonal. This gives us the lattice parameter, found using the below equation. (refer to W2 tut. work for practise) Lecture Four 8 Aug Continuing on Crystal Structures Calculating Atomic Packing in BCC - We can use the below equations to find the atomic packing factor: - Volume of atoms in one unit cell / volume of unit cell - See the below example: Explanation of example: - We know that in BBC there are 2 atoms per unit cell, therefore n = 2 4 3 - We can find the volume of an atom using 3 π𝑟 3 4𝑅 3 - Volume of unit cell = 𝑎 , therefore = ( ) 3 3π - Using our knowledge of volumes, we can derive the alternative APF equation: 8 FCC (Face centered cubic) Crystal Structure - In an FCC, the atoms touch each other along the diagonal of a face. - We can find n of FCC by doing the below:  - To find the lattice parameter (a) of a FCC we can use: - We can find the atomic packing factor of a FCC unit cell, by doing the following: Theoretical Density (ρ) - Tells us how the crystal structure is related to the density of a material Density = ρ = mass of atoms in unit cell / total volume of unit cell - Equation to help us calculate this:  𝑛𝐴 - Equation comes from mass of atom in unit cell is being 𝑁𝑎 , and total volume being 𝑉𝑐 Densities of Material Classes - In general P metals > P ceramics > P polymers - Metals have the biggest density because the have close-packing due to metallic bonding and often have large atomic masses - Ceramics have less dense packing and often have large atomic masses - Polymers have low packing density (often amorphous) and are lighter elements - Composites have intermediate values Practice Questions (Calculation) 1. Find theoretical mass ρ for FCC metal Copper: 2. Calculate the atomic packing factor of a simple cubic cell:  3. Mo crystallizes in a BCC structure, its atomic radius is equal to 0.136 nm and its atomic weight is 95.94 g/mol. Calculate the theoretical density. 𝐹𝑖𝑛𝑑𝑖𝑛𝑔 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑚𝑜𝑙𝑦𝑏𝑑𝑒𝑛𝑢𝑚 𝑖𝑛 𝐵𝐶𝐶 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒. 𝑛𝐴 1 4𝑅 𝑝 = 𝑁𝐴 * 𝑉𝐶 , 𝑎 = , 𝑛= 2 3 3 3 64𝑅 𝑉𝑒 = 𝑎 = 3 3 2 * 94.95 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑝 = 64 −7 3 23 ( )*(0.0136*10 ) *(6.023*10 ) 3 3 3 = 10. 28 𝑔/𝑐𝑚 3 𝑇ℎ𝑖𝑠 𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 𝑠𝑖𝑚𝑖𝑙𝑎𝑟 𝑡𝑜 𝑡ℎ𝑒 𝑒𝑥𝑝 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 (10. 22 𝑔/𝑐𝑚 ) Lecture Five 13 Aug Polymorphism or Allotropy Definition - Metals exist in more than one crystalline form; this is called polymorphism or allotropy - Temperature and pressure leads to change in crystalline forms (rearrangement of atoms) - Example: Iron exists in both BCC and FCC form depending on the temperature (a face transformation)  - As you heat up a material it expands meaning that the lattice parameter will also change; in the example able, this tells us that the different Irons will have different lattice parameters Point Coordinates Definition - Provide points for the cell eg origin is - Point coordinates for unit cell center are a/2, b/2, c/2 - Point coordinates for unit cell corner are 111, as seen below - Translation: integer multiple of lattice constants -> identical positions in another unit cell Crystallographic Directions - Algorithm - Vector repositioned if necessary to pass through origin - Read off projections in terms of unit cell dimensions a,b and c - Adjust to smallest integer values - Enclose in square brackets, no commas; [abc] - Rules - Don't use fractions, always integers/decimals eg [1 0 ½] will be changed to [2 0 1] - Don’t write negatives, instead use bar above negative number - Family of directions () represents all the directions that have the same properties; always 6 equivalent directions - For example : [1 0 0] is family to [0 1 0], [0 0 1] and then all with bar 1 Linear Density Definition - Solve density making use of coordinates - Equations for linear density of atoms (LD): Practice Question on Linear Density Determine the linear density of the following directions in BCC unit Cell if the material is iron with R = 0.124 nm. 1) Direction 100 2) Direction 110 3) Direction 111 Crystallographic Planes Definition - Just as we have family of directions, we have a family of planes - All planes in a family are parallel to each other and therefore have the same directions Miller indices - Represent the coordinate of the plane; reciprocals of the three axial intercepts for a plane, clear of fractions and common multiples - All parallel planes have the same miller indices - In cubic squares, the miller indices is the same as the vector quantity that represents the normal to the plane - Family of plane can be represented by {abd} - Algorithm as seen below: - Read off intercepts of plane with axes in terms of a, b, c - Take reciprocals of intercepts (inverse of.. 1/a) - Reduce to smallest integer values - Enclose in parentheses, no commas ie (hkl) Examples of Miller Indices Planar Density - The number of atoms centered on a plane, divided by the area of the plane - Needs to be exactly in the center; this is because of nucleus weight density etc etc Practice: Crystallographic Planes - We want to examine the atomic packing of crystallographic planes - In this example, we are looking at iron foil that can be used as a catalyst (the atomic packing of the exposed planes is important) 1. Draw (100) crystallographic planes for Fe (iron) and calculate the planar density 2. Draw (111) crystallographic planes for Fe (iron) and calculate the planar density X-Ray Diffraction - Technique used to measure the crystallographic - We use x-rays as our source of energy due to their frequency and wavelength being similar to the objects we want to measure in this case - Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation - Can’t resolve spacings smaller than wavelength - Spacing is the distance between parallel planes of atoms - Incoming x-rays diffract from the crystal planes (reflections must be in phase for a detectable signal) - Measurements of critical angle, theta, allows computation of planar spacing, d - This gives us bragg's law, which is the equation for distance (spacing between the plane/interplanar distance) and the angle of refraction (n is order of diffraction, always 1 here, and lander is wavelength):  - Example of a diffraction pattern below: - Diffraction seen as intensities/peaks collected 45, 65 and 82.5 - Each peak has corresponding plane coordinates eg 110, 200, 211 - Such values will allow us to find the interspacing distance between my planes (see below) Example using X-Ray Peak Diffraction - Using the first peak in the previous diffraction diagram, calculate the lattice parameter (a) for BCC iron - Hint: For cubic meters we have the following relation between the interplanar spacing (d) and lattice parameter (a): ` Amorphous Materials - Random spatial positions of atoms - Polymers: Secondary bonds do not allow formation of parallel and tightly packed chains during solidification, therefore, polymers can be semicrystalline 4− - Glass is a ceramic made up of 𝑆𝑖𝑂4 tetrahedron subunits (limited mobility) 8 - Rapid cooling of metals (10 𝐾/𝑠) can give rise to an amorphous structure (metallic glass) - Metallic glass has superior metallic properties Summary from Lecture Quick facts: - Atoms may assemble into crystalline or amorphous structures. - Common metallic crystal structures are FCC, BCC, and HCP.Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures. - We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). - Crystallographic points, directions and planes are specified in terms of indexing schemes. Crystallographic directions and planes are related to atomic linear densities and planar densities. Lecture Six 20 Aug Crystals as Building Blocks Single Crystal Materials - Some engineering applications require single crystals: - In a single crystal material, all the unit cells are in the same orientation relative to the reference you are looking at (x,y,z reference) - Anisotropic by defintion - This is not easily obtainable, more commonly you get polycrystalline materials (see below) - Properties of crystalline materials often are related to crystal structure: - For example: Quartz fractures more easily along some crystal planes than others  - Application of single materials: - Used within modern turbine blades (Ni Based Super Alloy) - Before they used single crystal materials in turbines, they used directionally solidified structure (columns) to reduce grain boundaries and put material crystals in the direction that faces the most load - This was developed further with single crystal were there is a single lattice orientation every way; therefore no grain materials - This is very expensive and hard to source as to why it isn’t used in many other applications outside of turbines Polycrystalline Materials - More common (naturally occurring) than single crystal structures - In polycrystalline materials, unit cells have different orientations over different areas - Examples of polycrystalline materials include: - Different coloured regions are determined by grain scales; in one grain unit cells are at same orientation, in different grains different orientations of latices (This seen in the different coloured section in the photos above) Single vs Polycrystals Materials - Single Crystals: - Properties vary with direction: anisotropic - Example: the modulus of elasticity (E) in BCC iron: - Polycrystals: - Properties may/may not vary with direction - In each grain (sections seen below) there is a different orientation of lattices - Because of the amount of grains in a material, there is a very high probability that all the different possible orientations of lattice are somewhere within the material (all the orientations are represented somewhere)  - If grains are randomly oriented, isotropic (f.1 below) - If grains are textured, anisotropic (f.2 below) f.1 f.2 - Throughout course, we can presume all the materials are isotropic unless stated otherwise Solidification of Metals Introduction - How the grains change when we move from liquid to solid - Metals are melted to produce finished and semi-finished parts - Two steps to solidification: - Nucleation: Formation of stable nuclei (needed to grow crystals) - Growth of nuclei: Formation of grain structure - Grain boundaries occur when two crystals meet each other and they can no longer grow anymore (solid meets solid, no more liquid) - Thermal gradients define the shape of each grain Formation of Stable Nuclei: Homogeneous vs Heterogneous - Two main mechanisms: Homogeneous (everything the same) and heterogeneous (something different in the system) - Homogeneous nucleation: - First and simplest case - Metals itself will provide atoms to form nuclei - Metal, when significantly undercooled has several slow moving atoms which bond each other to form nuclei; as material cools down, atoms slow down to a point where nuclei form - Cluster of atoms below critical size is called embryo  - If cluster of atoms reach critical size they grow into crystals (else get dissolved) - Cluster of atoms that are greater than critical size are called nucleus - As soon as the nucleus as stable, crystals will begin to grow - Heterogeneous Nucleation - Nucleation occurs in a liquid on the surface of structural material (eg insoluble impurities) - These structures called nucleating agents lower the free energy required to form stable nucleus - Example of nucleating agents is the mold in which holds the material that is being melted - this mold MUST be made from a different material to that being melted and MUST have a higher melting temperature - Nucleating agents also lower the critical size - Smaller amount of undercooling is required to solidify - Used excessively in industries Solidification in a Cast - Looking at the case of materials being solidified in a cast; casting is the process we use to produce unique shaped materials, pour liquid materials into a mold then solidify them by cooling (little control over the cooling process) - During solidification, cools faster on the edge so center parts will solidify last - Because of this grains at the center are smaller in size well center grains are elongated, this is because less stable nuclei can be created - Directions of inside, longer grains mimic the direction of heat flow during solidification - Grains can be: - Equiaxed (roughly same size in all directions) Or - Columnar (elongated grains) - Grain refiner - helps creates more stable nuclei, added to make smaller, more uniform, equiaxed grains (particular important for inner grains), gives more control