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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 Imperfections of Solids: Point Defects Definition of Imperfections of Solids in General - There is no perfect crystal; imperfections control a lot of the crystal properties Definition of Point Defects (Types of Imperfections) - Vacancy atoms (vacancies) - Vacant atomic sites in structures - Empty spaces where atom should have been located (lack of something) - Other atoms will have to tendency to want to fill this gap/free space meaning that instead of nice straight lines, we will see some lines become distorted - Most common point defect (most present in materials because the energy required to create a vacancy is much lower than the energy required to have an extra atom) - Interstitial atoms (self-interstitials) - “Extra” atoms positioned between atomic sites - When you have an extra atom that does not belong, and tries to make space for itself in what is meant to be a nicely packed atomic site - Substitutional atoms Calculating Equilibrium Concentration - Equilibrium Concentration of point defects - Equilibrium concentration varies with temperature - We can calculate this with an empirical equation (as seen below): - The above is an iranios equation; type of flow that describes the relationships between different parameters - Units MUST cancel out eg eV on top, K and -K on bottom  - Deriving different elements of the above equation: - Finding Qv (Activation energy) - Activation energy (Qv) is energy required to make defect (constants), dependant on material and type of defect - We can get Qv from experiment, where we measure the defect concentration, and replot the graph created to be ln Nv/N and 1/T and use the slope of the new line to get Qv - Finding k (Boltzmann’s constant) - The number you choose to use for Boltzmann's constant is dependant on the unit in which activation energy is in - Finding N (No. of potential defect sites) - Each lattice site is a potential vacancy/defect site, and therefore N represents number of atoms in material - Finding Nv (No. of defects) - See example below - Example of using equation to estimate vacancy concentration: - Answer: 3 𝑁𝐴 3 28 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑁 𝑓𝑜𝑟 1𝑚 : 𝑁 = 𝑝 * 𝐴𝐶𝑢 * 1𝑚 = 8. 0 * 10 𝑠𝑖𝑡𝑒𝑠 𝑚 𝑁𝑜. 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 * 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑜𝑛𝑒 𝑎𝑡𝑜𝑚 (𝐴/𝑁𝐴) 𝑅𝑒𝑐𝑎𝑙𝑙: 𝑝 (𝑑𝑒𝑛𝑠𝑖𝑡𝑦) = 𝑣 = 𝑉𝑜𝑙𝑢𝑚𝑒 (𝑉) −4 28 25 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑁𝑉 = (2. 7 * 10 )(8. 0 * 10 ) 𝑠𝑖𝑡𝑒𝑠 = 2. 2 * 10 𝑣𝑎𝑐𝑎𝑛𝑐𝑖𝑒𝑠 Proving/Observing Vacancy Concentration - In order to prove that there are vacancy concentration, the following experiment was done:  - Low energy electron microscope view of (110) surface NiAl - All the lines we can see in the right image below, are planes. If we were to take a cross section of this we would see that each line represents the change in height of atoms (as seen below) - Increasing T (temp) causes surface area of island of atoms to grow (the middle section/circle/plane grows in size and also causes number of vacancies to increase - This is because the equilibrium vacancy concentration increases via atom motion from the crystal to the surface where they join the island Point Defects in Alloys - Two outcomes if impurity (B) added to host (A): - If atom A and B are same size (substitutional solid solution) some atoms B will replace atoms A, but if they are different sizes (interstitial solid solution) the atoms B will slot themself within field of A atoms - Example of what happens when you have an alloy: Solid solution of B in A plus particles of a new phase (usually for a larger amount of B) - Some atom B has taken position of Atom B, minor disturbance Conditions for Substitutional Solid Solution - Substitutional solid solution is where Atoms B replaces Atoms A in a material, our question is, how much of material B can be added to material A - We can qualitatively measure how easy it is to put atom B in the lattice of Atom A - This is calculated with four rules: W.Hume; Rothery Rule (ordered by importance) 1. Δr (atomic radius) < 15% 2. Materials to have similar electronic activity (similar electronegativities) ; ie. proximity in periodic table 3. Need to have same crystal structure for pure metals (eg, FCC and BCC etc); not impossible with different crystal structures but much easier when they are 4. Ideally same valence number (no. of valence electrons); If all else is equal (rules 1, 2 and 3), a metal will have a greater tendency to dissolve a metal of higher valence than one of lower valence Other Types of Imperfections - Line Defects (Dislocations) and - Area/Surface Defects (Grain boundaries) (to be discussed in later lectures) Lecture Seven 22 Aug Continuing on from Imperfection of Solids: Point Defects Example Questions; Applications of Hume-Rothery Rules (Solid Solutions) 1. Would you predict more Al or Cu to dissolve Ni? 2. More Al or Co in Zn? 3. More Ag or Pd in Ni? Solubility - Different solubility (percentage of material B that can be dissolved into material A) for elements in other metals - For example: Imperfections in Solids - Composition - Equate specification of composition: - Weight percent: - Atom percent: - Note: There are some relations to go from weight percent to atom percent Interstitial Solid Solution - Note: Hume-Rothery Rules discussed earlier is only valid for substitutional solid solution, NOT Interstitial Solid Solution - Solute atoms fit in between to voids (interstices, spare space) of solvent atoms - Solvent atoms in this case should be larger than solute atoms so that solute atoms can fit within voids (see image below, solute in red, solvent in blue) - Example: Between 912 and 1294ºC interstitial solid solution of carbon in y iron (FCC) is formed - A maximum of 2.8% of carbon can dissolve interstitially in iron (if you put more of that, you're going to create different (phases or faces???)  - Practice questions: - Would it be easier to have a carbon atom as interstitial defect in FCC or BCC iron? What are the consequences? - Answer: On the left is a BCC lattice, on the right is a FCC lattice. - In FCC lattice, the green dots are the atoms in interstitial position. These atoms are always at the same location/position. So we ask, what is the maximum radius of atoms we could put in those locations. - For BCC, we have different types of interstitials (black boxes versus white boxes). The white box is the largest space available, while the black boxes are smaller in size. - While FCC is better packed, less disturbing for the lattice to have an atom of carbon in the FCC compared to BCC - The full answer to this question will be answered in next week's tutorial (29th of August) Line Defects Dislocations Defintion - Line defects occur when atoms form into these different levels; if we were to take a cross section of the cobalt-nickel alloy seen below we would have somewhat of a structure like this. - Dislocation by definition are line defects; slip between crystal planes result when dislocations move producing permanent (plastic) deformation - Slip lines can be observed by optical microscopy - Dislocations are visible in electron micrographs  - Example: Schematic Zinc before deformation and after deformation (tensile elongation/slip steps - slip of material) Types of Dislocations - Linear defects (dislocations) - One-dimensional defects around which atoms are misaligned - Edge dislocation (seen by blue line in below image): - Extra half-plane of atoms inserted in a crystal structure - b is perpendicular to dislocation line - Screw dislocation: - Spiral planar ramp resulting from shear deformation - b is parallel to dislocation line Note: b is measure of lattice distortion Burgers Vector in Edge Dislocations - Burgers vector (b) measures the size of the defect (distance between start and end point) Motion of Edge Dislocation - Dislocation motion requires the successive bumping of a half plane of atoms (from left to right here) - Bonds across the slipping planes are broken and remand in succession - This motion can only occur when a load or force is applied to the material - Below is an atomic view of edge dislocation motion from left to right as a crystal is sheared - watch what happens as black line (edge dislocation line) moves through Screw Dislocation Edge, Screw and Mixed Dislocations - Most dislocations are mixed; they start screw, age, turn to edge then in the middle mix Lecture Eight 27 Aug Dislocations and Crystal Structures Introduction - Structure: Close-packed planes and directions are preferred - Pink circles represent the close-packed plane on bottom - Gray circles represent the close-packed plane on top - Comparison among crystal structures: - FCC: Many close-packed planes/directions - HCP: Only one plane, 3 directions (metals) - BCC: none - Specimens that were tensile tested; see below tensile direction of Mg (HCP) and Al (FCC):  - The motion of a dislocation is not random; occurs because of bonds being broken and created (causes a small shift of atom position) - BCC is none because there is not a high difference between the planar density and linear density Review: Family of Plane - Set of planes that follow the same coordinates in different directions - For example with FCC: - Four planes in 111 family {111} - 4 planes of higher planar density x 3 directions of higher linear density = 12 slip systems Elastic/Plastic Deformation - Elastic deformation evaluates almost directly to fractures - If deformation can’t move (small no. of slip systems) plastic deformation can’t occur and therefore a fracture will occur - A material with a large amount of slip systems is ductile while a material with a few or none will be brittle - This is relevant as it determines what type of processes you will use to create different shaped materials Planar Defects in Solids Definition - Also known as surface defects; it’s a two dimensional defect Grain Boundaries - Most common surface defects we see within metals is grain boundaries - Problems occur because the two crystals that interact at grain boundaries dont have the same crystal structure. Therefore, we get a region of disorder because the atoms are going to try fill the holes (in incorrect position) to make up for density - Angle of misalignment/misorientation is the different in angle of the two interacting crystal structures (See the below figure) - Grain boundaries is the joint between two crystal (occur when crystal reach maximum growth and they grow into each other):  - Regions between crystals - Transition from lattice of one region to that of the other - Slightly disordered - Low density in grain boundaries - High mobility - High diffusivity - High chemical reactivity Planar Defects in Solids - Twin boundaries (plane): - Essentially a reflection of atoms position across the twin plane - Spacial grains (grains within grains / grain split into family of grains) - Formed during heat treatment and deformation - Stacking faults: - For FCC metals an error is ABCABC packing sequence - One plane is missing, or one has been added (out of order) - Ex: ABCABABC instead of ABCABCABC - In the below figure (left) we see a line pop out from the center of the image; this is because at that location we have planes that are in the wrong position (plane of atoms is missing) Microscopy Techniques to Measure Defects Microscopy in General - What do we observe during microscopy: - Grain size and shape - Defects in the materials: stacking faults, dislocation, slip lines, twin and grain boundaries, etc… - Crystallographic orientation of the grains - Different phases and their composition - Resolution ranges of the three techniques: Optical Microscopy - Optical Microscopy (oldest and simplest method) - Use to light to “excite” material (interact with surface of the material), then observe the result (the reflection) - In other cases we can use polarized light, which better defines the different crystals and each crystal reacts differently to polarized light, producing different visible colors (highlighting grains) - Useful up to 2000X magnification - Polishing removes surface features; the surface should have a mirror like surface aspect and needs to be as flat as possible to get best possible reaction from light (will reflect directly back to microscope - Etching (chemical treatment involving acid) polishes and changes reflectance, depending on crystal orientation - When studying grain boundaries with a optical microscope:  - Are imperfections - More susceptible to etching - May be revealed as dark lines - Change in crystal orientation across boundary - At the point at which grain boundaries occur there is a small groove that redirects light causing no signal and dark lines to appear (see below) Grain Size Measurements using Microscopy - Using optical microscopy we get a ‘map’ or grains and can use this to measure - We do this by plotting lines into the picture and marking the intersects between the plotted lines and the grain boundaries - Then, using the scale bar we can measure these individual lengths - We can find the average number of grain boundaries in one line using the following equations:  - It is not necessary to put the lines in all the same direction, for example if grains were elongated Scanning Electron Microscopy (SEM) - Second method of microscopy - The SEM uses electrons instead of light to produce an image - Electron gun creates a beam of electrons, anode and magnetic lens control and directs the electron beam, the interaction with the beam to the material, creates a signal - Two types of signals/information that can be picked up: - Backscattered electrons; - Electrons enter material and are then expelled from it - High energy electrons originating in the electron beam that are reflected or back-scattered of the specimen interaction volume - Contrasted image showing regions of different chemical composition - Secondary electrons; - Electron beam causes atoms of the material to get excited and those atoms then expel from the material - Low energy electrons that originate within a few nanometers from the surface - Well defined 3 dimensional images Transmission Electron Microscopy (TEM) - Difference with SEM is the way electrons interact with the material; In TEM the electrons are transmitted through the material, instead of being reflected off - TEM requires more sample preparation, eg material needs to be very thin (nanometers) to reduce amount of material atoms and ensure minimal energy is lost so that electrons can move through - Advantages of this technique: - Very high resolution can be reached - Material features (and defects) that cannot be observed otherwise can be studies (eg dislocation can only be seen with TEM) - Drawbacks of this technique: - Sample preparation is tedious - Only a small area of the sample can be analyzed (is it representational of the entire material?) - Material defects shown on TEM: Diffusion in Solids Definition of Diffusion in all Matter - Diffusion is a process of which a matter is transported through another matter - Examples include: - Movement of smoke particles in air (Very fast) - Movement of dye in water (relatively slow) - Solid state reactions (Very restricted movement due to bonding - helps us control and change properties of a material) - While diffusion occurs in all matter, we are only focusing on diffusion in the solid state Kirkendall Effect - Kirkendall effect helped us first discover diffusion within solid materials - Discovered in 1942, the Kirkendall effect describes what happens when two solids diffuse into each other at different rates Interdiffusion (two materials) - Interdiffusion occurs in an alloy where atoms tend to migrate from regions of high concentration to regions of low concentration - The below diagram shows the interdiffusion of atoms over time: - Note we can see that interdiffusion is seen in the center, where boundaries still stay in pure/original state - If we have the same flux of atom B and A, the intercept would be 50/50 right at the original boundary point Self diffusion (one material) - Self diffusion occurs in an elemental solid, where atoms can also migrate (move throughout material over time) Diffusion Mechanisms in General - There are three hypostases for atomic diffusion mechanism (refer to figures below): (a) A direct exchange mechanism (switch) - Although this seems simple, the movement of the two as they swap places will require surrounding atoms to move to allow space for them to swap, meaning it is difficult and requires lots of energy (b) Ring mechanism (rotation of four atoms) - Less energy than part (a), but since four atoms need to be moved, it still requires a lot of energy (c) Vacancy mechanism - Takes much less energy than the others (still a bit as one atom still needs to move) meaning it is more probable to occur Vacancy Diffusion - Atoms exchange with vacancies - Applies to substitutional impurities atoms - Rate depends on: - The number of vacancies - Activation energy to exchange (different for different atoms) Diffusion Simulation - Simulation of interdiffusion across an interface - Rate of substitutional diffusion depends on: - Vacancy concentration - Frequency of jumping - Watch below how the intersection line changes overtime: Interstitial Diffusion - Smaller atoms can diffuse between atoms Question Time - What mechanism is the fastest: vacancy or interstitial diffusion? - Interstitial diffusion is faster because the diffusing atoms move through the small spaces (interstitial sites) between the host atoms in the lattice, which are typically unoccupied. These interstitial sites are more abundant and readily accessible, allowing the diffusing atoms to move more easily and quickly. - In contrast, vacancy diffusion relies on the presence of vacant lattice sites (vacancies) for atoms to move from one position to another which is less probable. Since vacancies are less common and require the energy to create or migrate, vacancy diffusion is generally slower. Application of Diffusion Case Hardening - Diffuse carbon atoms into host iron atoms at surface - Example of interstitial diffusion in a case hardened heart - Result: The presence of C atoms makes iron (steel harder) - This is useful within gearbox as the constant interaction with other materials when spinning mean that it is prone to erosion/weathering - increasing hardness makes it more resistant to this erosion (note: we do not want to make it more brittle as then it will just snap)