1133 lines
36 KiB
Markdown
Executable File
1133 lines
36 KiB
Markdown
Executable File
---
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author: Alvie Rahman
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date: \today
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title: MMME1029 // Materials
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tags: [ uni, nottingham, mechanical, engineering, mmme1029, materials ]
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---
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\tableofcontents
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# Lecture 1 (2021-10-04)
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## 1A Reading Notes
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### Classification of Energy-Related Materials
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- Passive materials---do not take part in energy conversion e.g. structures in pipelines, turbine
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blades, oil drills
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- Active materials---directly take part in energy conversion e.g. solar cells, batteries, catalysts,
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superconducting magnests
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- The material and chemical problems for conventional energy systems are mostly well understood and
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usually associated wit structural and mechanical properties or long standing chemical effects like
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corrosion:
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- fossil fuels
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- hydroelectric
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- oil from shale and tar
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- sands
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- coal gasification
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- liquefaction
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- geothermal energy
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- wind power
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- bomass conversion
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- solar cells
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- nuclear reactors
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### Applications of Energy-Related Materials
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#### High Temperature Materials (and Theoretical Thermodynamic Efficiency)
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- Thermodynamics indicated that the higher the temperature, the greater the efficiency of heat to
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work:
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$$ \frac{ T_{high} - T_{low} }{ T_{high} } $$ where $T$ is in kelvin
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- The first steam engines were only 1% efficient, while modern steam engines are 35% efficient
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primarily due to improved high-temperature materials.
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- Early engines made from cast iron while modern engines made from alloys containing nickel,
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molybdenum, chromium, and silicon, which don't fail at temperature above 540 \textdegree{}C
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- Modern combustion engines are nearing the limits of metals so new materials that can function
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at even higher temperatures must be found--- particularly intermetallic compounds and ceramics are
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being developed
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## Types of Stainless Steel
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- Type 304---common; iron, carbon, nickel, and chromium
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- Type 316---expensive; iron, carbon, chromium, nickel, molybdenum
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## Self Quiz 1
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1. What is made of billion year old carbon + water + sprinkling of stardust?
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> Me
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2. What are the main classifications of materials?
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> Metals, glass and ceramics, ~~plastics, elastomers,~~ polymers, composites, and semiconductors
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3. [There are] Few Iron Age artefacts left. Why?
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> They rusted away
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4. What is maens by 'the micro-structure of a material'?
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> The very small scale structure of a material which can have strong influence on its physical
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> properties like toughness and ductility and corrosion resistance
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5. What is a 'micrograph' of a material?
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> A picture taken through a microscope
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6. What microscope is used to investage the microstructure of a material down to a 1 micron scale
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resolution?
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> Optical Microscope
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7. What microscope is used [to investigate] the microstructure of a material down to a 100 nm scale
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resolution?
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> Scanning Electron Microscope
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8. What length scales did you see in the first slide set?
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> 1 mm, 0.5 mm, 1.5 \textmu{}m
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9. What material properties were mentioned in the first slide set?
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> Hardness, brittleness, melting point, corrosion, density, thermal insulation
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## Self Quiz 2
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1. What is the effect of lowering the temperature of rubber?
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> Makes it more brittle, much less elastic and flexible
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2. What material properties were mentioned in the second slide set?
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> Young's modulus, specific heat, coefficient of thermal expansion
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# Lecture 2
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## Properties of the Classes
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### Metals
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- Ductile (yields before fracture)
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- High UFS (Ultimate Fracture Stress) in tension and compression
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- Hard
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- Tough
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- High melting point
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- High electric and thermal conductivity
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### Ceramics and Glasses
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- Brittle --- elastic to failure, no yield
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- Hard (harder than metals)
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- Low UFS under tension
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- High UFS under compression
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- Not tough
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- High melting points
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- Do not burn as oxide ceramics are already oxides
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- Chemically resistant
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- Poor thermal and electric conductivity
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- Wide range of magnetic and dielectric behaviours
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### Polymers
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- Organic---as in organic chemistry (i.e. carbon based)
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- Ductile
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- Low UFS in tension and compression
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- Not hard
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- Reasonably tough
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- Low threshold temperature to charring and combustion in air or pure oxygen
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- Low electrical and thermal conductivity
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- There are some electrically conductive polymers
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### Composites
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- Composed of 2 or more materials on any scale from atomic to mm scale to produce properties that
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cannot be obtained in a single material
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- Larger scale mixes of materials may be called 'multimaterial'
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- Material propertes depends on what its made of
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## Terms
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### Organic vs Inorganic Materials
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- Organic materials are carbon based
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- From chemistry, organic compounds are ones with a C-H bond
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- Inorganic compounds do not contain the C-H bond
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### Crystalline vs Non-Crystalline Materials
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- Most things are crystalline
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- Ice
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- Sugar
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- Salt
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- Metals
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- Ceramics
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- Glasses are non-crystalline
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## Material Properties
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### Density
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$$\rho = \frac m v$$
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- Density if quoted at STP (standard temperature and pressure---$298$ K and $1.013\times 10^5$ Pa)
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- Metals, ceramics, and glasses are high density materials
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- Polymers are low density
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- Composites span a wide range of density as it depends on the materials it is composed of
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e.g. composites with a metal matrix will have a much higher density than those with a polymer
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matrix
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### Melting Points
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- Measured at standard pressure and in an intert atmosphere (e.g. with Nitrogen, Argon, etc)
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- Diamond and graphite will survive up to 4000 \textdegree{}C in an inert atmosphere but would
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burn at around 1000 \textdegree{}C in oxygen
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- High melting points -> high chemical bond strength
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### Corrosion
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- It's not just metals that corrode
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- Polymers
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- UV degradation
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- Water absorption can occur in degraded polymers
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- Glass
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- Leaching
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- Sodium ions can leave the glass when covered in water. If the water stays, the high pH water
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can damage the class
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## Self Quiz
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### Consolidation Questions 1
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1.
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i. Metal
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ii. Titanium
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2.
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i. Polymer
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ii. Polyester
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3.
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i. Ceramic
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ii. Alumino-silicate
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4.
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i. Composite
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ii. GFRP or CFRP
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5.
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i. Metal
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ii. Aluiminium alloy
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6.
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i. Metal
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ii. Aluiminium alloy
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7.
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i. Polymer
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ii. Acrylic
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8.
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i. Ceramimc
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ii. Glass
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9.
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i. Composite
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ii. Concrete
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### Consolidation Questions 2
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> ~~C~~ B
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# Polymers
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## Introduction to Polymers
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There are 3 types of polymers:
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- thermoplastics
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No cross links between chains.
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The lack of cross links allows recycling of polymers by heating it above the glass transition
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material, $T_g$, lowering the viscosity.
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An example of thermoplastics is PET, used in water bottles
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- thermosets
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has lots of cross-links between chains, making it more rigid.
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Heating does not lower its viscosity making them much harder/impossible to recycle.
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and example of thermosets is melamine formaldehyde, used on kitchen tabletops
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- elastomers
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has some cross links and a lot of folding of chains
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Latex is an example of an elastomer
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Polymers are relatively new materials, lightweight, durable, flammable, and degraded by UV light.
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They are made of long carbon-carbon chains.
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### Stress-Strain Curve of Polymers
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## Thermoplastics
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The simplest polymer is poly(ethene):
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When 2 polymer chains get close together, Van der Waals (vdw) forces keep them together.
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vdw forces are very weak, much weaker than the covalent bonds inside the polymer.
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### Stress Strain Curve
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- During linear deformation, the carbon chains are strethed.
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- At yield stress, the carbon chains get untangled and slide past eache other.
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- Necking initially allows the chains to slide at lower stress.
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- As the chains pull, align, and get closer, the vdw forces get stronger and more stress is required
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to fracture.
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### Crystalline and Amorphous/Glassy Solids (Heating and Cooling)
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#### Amorphous Thermoplastics
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- As you heat above $T_g$, the chains get easier to move past each other.
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- It is known as an *amorphous supercooled liquid*.
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- There is not really a melting point are there are no crystals, but $T_m$ is the point where the
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chains are easy to move
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#### Crystalline Polymers
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- The glass transition point does not exist for crystalline polymers
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- The solid is difficult to deform below $T_m$ and is not ductile
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- Above $T_m$ the chains are very easy to move past each other
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#### Semi-Crystalline
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- Below $T_g$, only local movements in chains are possible, so the material is less ductile.
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The solid crystalline regions makes it difficult to move the chains.
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- Between $T_g$ and $T_m$, the glassy chains are easier to move but the crystalline regions remain
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difficult
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- Above $T_m$ the chains easily move past each other
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### Specific Volume vs Temperature
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#### Path ABCD
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- a-b --- Start cooling the true liquid
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- b-c --- At the freezing point, $T_m$, the true liquid freezes diretly to a crystalline solid
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- c-d --- The crystalline solid cools t room temperature as the temperature is lowered
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#### Path ABEF
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- a-b --- start cooling the true liquid
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- b --- at the freezing point nothing freezes
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- b-e --- the liquid becomes *supercooled* and contracts and becomes more viscous as the temperature
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decreases.
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The supercooled liquid region is between $T_g$ and $T_m$
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Supercooling requires you to cool the sample quicker than you would for path ABCD
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- e --- $T_g$ is reached and the supercooled liquid sets to a amorphous solid
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- e-f --- the amorphous solid cools from room temperature and contract as the temperature is lowered
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## Relative Molar Mass and Degree of Polymerisation
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- Number Average RMM --- $\bar M_n = \sum x_iM_i$
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- Weight Average RMM --- $\bar M_w = \sum w_iM_i$
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- Degree of polymerisation --- $n_n = \frac {\bar M_n} m$ and $n_w = \frac {\bar M_w} m$
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where
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- $M_i$ is the RMM of the chain
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- $x_i$ is the fraction of the polymer that is composed of that chain by number/quantity
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- $w_i$ is the fraction of the polymer that is composed of that chain by mass/weight
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- $m$ is the RMM of the monomer from which the polymer was made
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## Making Polymers
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There are two ways to make polymers:
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- [Addition Poymerisation](http://www.chemguide.co.uk/14to16/organic/addpolymers.html)
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- [Condensation Polymerisation](https://www.chemguide.uk/14to16/organic/condpolymers.html)
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# Elastic Deformaion
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Elastic deformation is deformation where the material will return to original shape after the
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applied stresses are removed.
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Elastic deformation is the first type of deformation that happens when stresses are applied to
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a material and is represented by the straight line at the beginning of a stress-strain curve.
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## Modulus of Resillience ($E_r$)
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This is the area under the elastic portion of a stress-strain graph of a material.
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# Plastic Deformation
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## Toughness (Absorbing Energy Through Plastic Deformation)
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- The toughness of a material is its ability to absorb energy through plastic deformation
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without fracturing
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- The material toughness of a ductile material can be determined by finding the area under its
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stress-strain curve (e.g. by integrating the graph)
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- Brittle materials like ceramics and glasses exhibit no material toughness
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- Ductile materials have a possibility of achieving large material toughness
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Ductility measures how much something deforms plastically before fracture, but just because a
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material is ductile does not make it tough.
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*The key to high material toughness is a good combination of large ultimate fracture stress and
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large ductility*.
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- The unit of toughness is energy per unit volume as toughness can be mathematically expressed as:
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$$toughness = \int^{\varepsilon_f}_0\! \sigma \,\mathrm{d}\varepsilon
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= \frac{\text{Energy}}{\text{Volume}} $$
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- A metal may have satisfactory toughness under static loads but fail under dynamic loads or impact
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This may be caused by the fact that ductility and toughness usually decrease as rate of loading
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increases.
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- Ductility and toughness decreasee with temperature
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- Notches in the material affect the distribution of stress in the material, potentially changing
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it from a uniaxial stress to multiaxial stress
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### Charpy Impact Test
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Measures material toughness by determining the amount of energy absorbed during fracture.
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It works by essentially dropping a hammer into a sample whose dimensions are standardized
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(usually either by BSI or ISO) and measuring how high the hammer goes up on the other side,
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after it breaks the material
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The height of the hammer after impact will tell you how much enery is left in it, and therefore
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how much has been aborbed by the now broken sample.
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Under a microscope, more ductile fractures appear fibrous or dull, whereas less ductile surfaces
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have granular or shiny surface texture.A
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The charpy test has a couple issues:
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- Results are prone to scatter as it is difficult to achieve a perfectly shaped notch
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- Temperature has to be strictly controlled since it affects a material's ductility
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#### The setup of a charpy impact test
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1. Sample is made to standardized dimensions, with a notch
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2. Sample is placed on support
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3. A very heavy hammer pendulum of mass $m$ is dropped from rest at $h_0$ to swing about a pivot,
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reaching $E_{kmax}$ vertically below the pivot.
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4.
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a. If no sample is in place then the hammer will swing back up on the other side to a height of
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$h_h$ where theoretically $h_h = h_0$
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b. With a sample placed vertically below, some of the $E_k$ is transferred to the sample to bend
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and (usually) break the sample.
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If breaks the sample, it will swing up to the other side, where its max height, $h_f$ can be
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used to calculate how much energy was used to break the sample:
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$$E = mg(h_h-h_f)$$
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Where $g$ is acceleration due to gravity.
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# Ductility
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Ductility is the plastic deformation a material withstands before fracture.
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# Griffith Surface Flaws
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These flaws vary in size and shape.
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They limit the ability of any material, brittle or ductile, to withstand tensile stresses as they
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concentrate the tensile forces applied to a smaller area.
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The stress at the tip of the flaw:
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$$\sigma_{actual} = 2\sigma\sqrt{\frac a r}$$
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For deep ($a$ is large) or thin ($r$ is small) the stress is magnified and, if it exceeds the UFS
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in a brittle material, the flaw will grow into a crack, resulting in the brittle material
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fracturing.
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However in a ductile material, the tip of the flaw can heal, reducing $a$ and increasing $r$.
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This is due to the chemical structure of ductile materials like metals.
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## Stress Intensity Factor
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Stress Intesity Factor, $K$:
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$$K = f\sigma\sqrt{\pi a}$$
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where:
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- $f$ is the geometry factor (1 would represent an infinite width sample, and 0 a 0 width sample)
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- $\sigma$ is applied tensile strength
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- $a$ is flaw depth
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## Fracture Toughness
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![An example sample for testing fracture toughenss. From: <https://www.researchgate.net/figure/Compact-tension-sample-geometry-used-for-fracture-toughness-measurement_fig2_340037774> [accessed 8 Nov, 2021]](./images/Compact-tension-sample-geometry-used-for-fracture-toughness-measurement.png)
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The value of $K$ that causes the notch to grow and cause fractures.
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This is value is known as the fracture toughness, $K_c$.
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At low thicknesses fracture toughness depends on thickness but as thickness increases, $K_c$
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decreases to the constant value, the plane strain fracture toughness, $K_{1c}$.
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# Composites
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Composites are made of two or more materials, which when combined together, at up to a milimetre
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scale, have superior properties to their parent materials.
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Composites tend to be 2-phase: a dispersed phase in a matrix.
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The disepersed phase tends to be fibres (large aspect ratio) or particles (low aspect ratio) which
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are embedded in a matrix, which are often resins.
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Composite properites are affected by the dispersed phase geometry:
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- Shape
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- Size
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- Distribution
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- Relative orientation (for fibres)
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## Rule of Mixtures
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$E_c$ lies between the arithmetic mean (upper limit):
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$$V_mE_m + E_pV_p$$
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and the geometric mean (lower limit):
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$$\frac{V_mE_mE_pV_p}{V_mE_m + E_pV_p}$$
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Where $E_c$, $E_m$, $E_p$ are the Young's moduluses of the composite, matrix, and particles,
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respectively, and $V_m$ and $V_p$ are the volume of the matrix and particles, respectively.
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## Particle Reinforced Composites
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### Applications of Composites
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<details>
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<summary>
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#### Tungsten Carbide Cobalt for Cutting Tools
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</summary>
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The Tungest Carbide (WC) particle are a truly brittle ceramic.
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They are very hard but the brittleness means they are easy to break.
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The solution is to hold small WC particles in a ducitle metal matrix.
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In this case it is Cobalt (Co).
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This way, crack in one WC particle does not necessarily mean other particles are broken,
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meaning the cutting tool overall still works.
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Another advantage of this composite is that WC is not very thermally conductive and has a high
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melting point, which allows it to work well the environment it's in.
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</details>
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<details>
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<summary>
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#### Resin Bonded Alumina for Sanding Disks
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</summary>
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This is another example of brittle but hard ceramics being put in a ductile matrix.
|
|
In this case it's a resin.
|
|
|
|
It follows the same idea---separating the ceramics into small particles means the particles can
|
|
break and the product still works overall, as there are thousands of particles which are not broken.
|
|
</details>
|
|
|
|
## Fibre Reinforced Composites
|
|
|
|
### Specific Property
|
|
|
|
Specific Property of a composite is a property divided by density of composite.
|
|
Here are some examples of specific properties:
|
|
|
|
- Specific ultimate tensile stress $= \frac{\sigma_{UTS}}{\rho_c}$
|
|
- Specific Young's modulus/stiffness $= \frac{E_c}{\rho_c}$
|
|
|
|
### Influence of the Fibres
|
|
|
|
Depends on:
|
|
|
|
- Fibre type
|
|
- Fibre length and diameter
|
|
- Fibre orientation
|
|
- Strength of bond between fibre and matrix
|
|
|
|
### Stress Strain Graph of a Fibre Reinforced Composite
|
|
|
|
|
|
|
|
Note that the composite fails at the same strain as the fibres but yields at the same strain as
|
|
the polymer matrix.
|
|
|
|
The elastic behaviour of the composite before yielding is dependent on the strength of the chemical
|
|
bonds between the surface of the fibre and matrix.
|
|
|
|
### Mechanical Performance of a Fibre Reinforced Composite
|
|
|
|
- Stress/strain behaviour of fibre
|
|
- Stress/strain behaviour of matrix
|
|
- Fibre volume fraction
|
|
- Applied stress direction
|
|
|
|
Longitudinal is along direction of fibres, transverse is 90\textdegree to direction.
|
|
|
|
Fibre composites tend to be much much weaker in transverse direction:
|
|
|
|
Composite | Longitudinal UTS | Transverse UTS
|
|
------------ | ---------------- | --------------
|
|
GF/PET | 700 | 20
|
|
CF/Epoxy | 1000 | 35
|
|
Kevlar/Epoxy | 1200 | 20
|
|
|
|
(All units in MPa)
|
|
|
|
# Thermal Properties of Materials
|
|
|
|
## Specific Heat Capacity
|
|
|
|
How much heat energy is required to raise the temperature of a body by one unit:
|
|
|
|
$$ C_p = \frac{\Delta E}{m\Delta T} $$
|
|
|
|
where $c$ is specific heat capacity.
|
|
|
|
It is measured at a constant pressure, usually $1.013\times 10^5$ Pa.
|
|
|
|
## Molar Heat Capacity
|
|
|
|
$$C_pm = \frac{\Delta E}{n\Delta T}$$
|
|
|
|
<details>
|
|
<summary>
|
|
|
|
#### What is a mole?
|
|
|
|
</summary>
|
|
|
|
> The mole (symbol: mol) is the base unit of amount of substance in the International System of
|
|
> Units (SI).
|
|
> It is defined as exactly $6.02214076\times 10^{23}$ elementary entities ("particles")
|
|
|
|
~ [Wikipedia: Mole (unit)](https://en.wikipedia.org/wiki/Mole_(unit))
|
|
|
|
</details>
|
|
|
|
<details>
|
|
<summary>
|
|
|
|
#### How Much Does a mol of Something weigh?
|
|
|
|
</summary>
|
|
|
|
A mol of an element weighs its relative atomic mass ($A_r$) but in grams.
|
|
For example, Carbon-12 has an $A_r$ of 12 (as it's made of 6 neutrons, 6 protons, and 6 electrons
|
|
which have negligible mass) so a mol of Carbon-12 has a mass of 12 g.
|
|
|
|
</details>
|
|
|
|
## Thermal Expansion
|
|
|
|
<details>
|
|
<summary>
|
|
|
|
### Origin of Thermal Expansion
|
|
|
|
</summary>
|
|
|
|
All atomic bonds vibrate, on the magnitude of gigahertz.
|
|
The bonds vibrate about a mean positoin and the vibration is a simple harmonic motion.
|
|
|
|
From the graph below you can see that as energy (in the form of heat) is supplied to the bonds,
|
|
the amplitude of the vibrations get larger and larger.
|
|
You can also see the mean position of the bond gets further and further away, meaning the volume
|
|
of the material also is increasing.
|
|
The mean position of the bond is what dictates the volume, as this means the inter-atomic
|
|
separation increases.
|
|
|
|
|
|
|
|
Morse potential is the energy well between 2 bonded atoms.
|
|
The graph is asymmetric due to the repulsion experienced by atoms as they apporach.
|
|
|
|
</details>
|
|
|
|
### Linear Coefficient of Thermal Expansion
|
|
|
|
$$\alpha_L = \frac{\Delta L}{L_0 \Delta T}$$
|
|
|
|
where $L$ is the sample length.
|
|
|
|
<details>
|
|
<summary>
|
|
|
|
#### Example 1
|
|
|
|
A 1 m long bar of aluminium metal cools in the solid state from 660 \textdegree{}C to
|
|
25 \textdegree{}C.
|
|
Calculate the length of the bar after it cools down, given $\alpha_L = 25\times10^{-6}$ K$^{-1}$.
|
|
|
|
</summary>
|
|
|
|
\begin{align*}
|
|
l_0 &= 1 \\
|
|
\Delta T &= T_f - T_0 = 25 - 660 = -635 \\
|
|
\\
|
|
\alpha_L &= \frac{l_f - l_0}{l_0 \Delta T} \\
|
|
\alpha_L l_0 \Delta T &= l_f - l_0 \\
|
|
l_f &= \alpha_L l_0 \Delta T + l_0 = 0.984
|
|
\end{align*}
|
|
|
|
</details>
|
|
|
|
### Linear Thermal Expansion and Isotropism
|
|
|
|
Since isotropic solids have the same properties in all directions, you can say that for an
|
|
isotropic solid:
|
|
|
|
$$\alpha_V = 3\alpha_L = \frac{\Delta V}{V_0 \Delta T}$$
|
|
|
|
### Reasons to Care About Thermal Expansion
|
|
|
|
- A coating on a material may fail if the thermal expansion coefficients do not match
|
|
- A brittle material may thermally shock and fracture due to thermal expansion mismatch between
|
|
the ouside and inside, especially if the material is not very thermally conductive
|
|
|
|
## Thermal Conductivity
|
|
|
|
Thermal conductivity is the rate at which heat power is transferred through a material.
|
|
|
|
$$\frac{Q}{A} = k \frac{\Delta T}{\Delta x}$$
|
|
|
|
where $Q$ is heat power, $A$ is area of the surface, $\frac{\Delta T}{\Delta x}$ is the
|
|
temperature gradient, and $k$ is the thermal conductivity constant.
|
|
|
|
<details>
|
|
<summary>
|
|
|
|
### Origin of Thermal Conductivity
|
|
|
|
</summary>
|
|
|
|
Heat is transferred through materials by electrons (and partially by atomic vibrations)
|
|
|
|
Metals have high thermal conductivity as their delocalised 'sea' electrons are about to move about
|
|
easily.
|
|
This makes them excellent conductors of heat and electricity.
|
|
|
|
Ceramics, glasses, and polymers do not have delocalised electrons and are therefore poor conductors
|
|
of heat and electricity (they are insulators).
|
|
|
|
Polymer foams are even better insulators because they have holes which lowers their density.
|
|
|
|
</details>
|
|
|
|
# Chemical Bonding of Materials
|
|
|
|
Chemical bonds are what holds a material together in solid state.
|
|
There are 5 main types of bonds:
|
|
|
|
Type | Dissociation energy
|
|
-------- | -------------------
|
|
Ionic | 600 to 1500
|
|
Covalent | 300 to 1200
|
|
Metallic | 100 to 800
|
|
Hydrogen | 4 to 23
|
|
vdw | 0.4 to 4
|
|
|
|
The dissociation energy is the energy required to break the bond, or the strength of the bond.
|
|
|
|
## Materials and their Properties and Bonding
|
|
|
|
### Ceramics and Glasses
|
|
|
|
Ceramics and glasses are composed of mixed ionic and covalent bonding.
|
|
Their strong and rigid bonds have no ability to slide past each other.
|
|
This makes the materials brittle.
|
|
|
|
### Metals
|
|
|
|
Metals are based on metallic bonding (woah).
|
|
This type of bonding *does* allow for ions to slide past each other, making metals ductile.
|
|
|
|
### Polymers
|
|
|
|
Polymer chains made of C-C covalent bonds are strong, like those found in ceramics.
|
|
|
|
However, in thermoplastics polymers, the materials can yield by having the chains untangle and
|
|
then align, as the chains slide past each other.
|
|
This means that **stronger bonds between polymer chains means a higher yield stress in thermoplastic
|
|
polymers**.
|
|
|
|
# Crystallisation of Materials
|
|
|
|
## Atomic Arrangement
|
|
|
|
- No order
|
|
- Short range order
|
|
|
|
Silica glasses have short range order on the atomic scale.
|
|
They are composed of regular SiO$_4$ units which all have the same bond length and bond angles.
|
|
|
|
However, these units bond together irregularly, which results in different length chemical bonds
|
|
and angles between the units, meaning they do not have any long range order.
|
|
|
|
- Long range order
|
|
|
|
## Cubic Unit Cells
|
|
|
|
- Lattice Parameter --- One side of a unit cell
|
|
|
|
The lattice parameter can be different for each side of a cell.
|
|
|
|
- Simple cubic unit (SC):
|
|
|
|
|
|
|
|
Lattice Parameter = 2r
|
|
|
|
- Face centred cubic (FCC)
|
|
|
|
|
|
|
|
- Body centred cubic (BCC)
|
|
|
|
|
|
|
|
### Packing Factor
|
|
|
|
$$\text{packing factor} = \frac{\text{ions per unit cell} \times V_{ion}}{V_{cell}}$$
|
|
|
|
### Theoretical Density
|
|
|
|
$$\text{theoretical density} = \frac{\text{ions per unit cell} \times m_{ion}}{V_{cell}}$$
|
|
|
|
### Polymorphism
|
|
|
|
Example of a polymorphic solid-state phase transfomration of iron at 1185 K and 1 atm:
|
|
|
|
$$\text{Fe}_{\text{BCC}} \longleftrightarrow \text{Fe}_{\text{FCC}}$$
|
|
|
|
Below 1185 K and at 1 atm, only BCC exists. Above 1185 K and at 1 atm, only FCC exists.
|
|
|
|
### Points, Directions, Planes in a Cubic Unit Cell
|
|
|
|
|
|
|
|
### Slip Systems in Metals
|
|
|
|
Metal ions lying in close-packed planes and directions move more easily, increasing ductility.
|
|
The combination of a close packed plane and direction is called a *slip system*.
|
|
|
|
A close packed direction is where ions touch all the way along the direction.
|
|
|
|
A close packed plane is where ions touch all the way on a plane.
|
|
|
|
FCC metal ductility is mainly controlled by the *(111) slip plane*
|
|
|
|
|
|
|
|
|
|
## X-Ray Diffraction (Bragg's Law)
|
|
|
|
The wavelength of x-rays, $\lambda$, is roughly equal to the distance, $d$, between atom/ion layers.
|
|
This allows x-rays to probe for $d$ via Bragg's Equation:
|
|
|
|
|
|
|
|
Requirements for the x-rays:
|
|
|
|
- Monochromatic
|
|
- Coherent (phase difference of $2\pi n$ where n is any integer)
|
|
- Parallel with each other
|
|
|
|
|
|
The incoming x-rays 1 and 2 strike the rows of ions in the crystal and are diffracted, which can be
|
|
considered reflection at the atomic level.
|
|
The angle of incidence equals the angle of reflection.
|
|
|
|
The outgoing x-rays 1 and 2 are coherent only if the extra path travelled by ray 2, $2d\sin\theta$
|
|
is any multiple, $n$, of $\lambda$. Or:
|
|
|
|
$$n\lambda = 2d\sin\theta$$
|
|
|
|
This is Bragg's Law.
|
|
|
|
# Metals
|
|
|
|
## Defects on the Atomic Scale
|
|
|
|
Defects on the atomic scale have a significant effect on yield stress, ultimate tensile stress, and
|
|
ultimate fracture stress.
|
|
|
|
The yield stress of a real metal(-alloy) is much lower than the theoretical yield stress for
|
|
the perfect metal(-alloy) crystal.
|
|
This difference is because of the defects in the metal, particularly dislocations, as the
|
|
dislocations allow the ions to slide past each other at much lower yield stresses.
|
|
|
|
The 5 types of defects are:
|
|
|
|
- Grain boundaries
|
|
- Vacancies (missing ion)
|
|
- Dislocations (missing row of ions)
|
|
- Impurity ions
|
|
- Crystalline includison
|
|
|
|
|
|
|
|
### Dislocation Movement vs. Simple Sliding
|
|
|
|
The layers of ions in a crystalline metal could simply over each other:
|
|
|
|
|
|
|
|
However, the stress required for simple sliding is much higher than the stress required to move a
|
|
dislocation.
|
|
This is because dislocation motion is successive sliding of the partial plane of ions under applied
|
|
shear stress (black arrow).
|
|
The vacancy in the slip plane (yellow arrow) moves in steps in sequence from left to right.
|
|
|
|
|
|
|
|
If there are no dislocations then plastic deformation is delayed to a higher applied stress,
|
|
meaning the yield stress of the metal would be much higher.
|
|
|
|
Dislocations move more easily on specific planes and in specific directions called the
|
|
slip planes and slip directions which make up what is known as the
|
|
[slip system](#slip-systems-in-metals).
|
|
|
|
There are a very large amount of dislocations in metals and alloys.
|
|
Dislocation density is expressed as total length of dislocations per unit volume.
|
|
|
|
## Single Crystal Metals
|
|
|
|
|
|
|
|
Normally when a molten metal is cooled to a solid, then lots of tiny crystals (grains) grow in
|
|
different directions until they impinge.
|
|
The grain boundaries are a source of mechanical weakness.
|
|
|
|
A single crystal metal is one for which the casting is cooled to form just one giant crystal:
|
|
|
|
1. The molten metal is cast into a mould
|
|
2. At the very base of the mould, the temperature is dropped and the alloy crystallises into many
|
|
little crystals
|
|
3. The crystals grow upwars through the liquid and meet a spiral tube and are constricted
|
|
4. This tube only allows one crystal to grow through the spiral and then into the main mould
|
|
|
|
## Polycrystalline Metals
|
|
|
|
Most normal metals you see everyday are polycrystalline.
|
|
|
|
|
|
|
|
|
|
|
|
## Elastic and Plastic Strain in Metals
|
|
|
|
When you apply a tensile stress to a mteal, this will produce a shear stress in any part of the
|
|
metallic lattice that is not parallel or perpendicular to the applied stress.
|
|
Under the action of shear stress, the metallic lattice will tend to experience a combination of
|
|
elastic strain and plastic strain:
|
|
|
|
|
|
|
|
## Raising the Yield Stress of a Metal
|
|
|
|
There are 4 main ways to raise the yield stress of a metal:
|
|
|
|
- Make a solid-solution---by metal alloying or atomic addition
|
|
- Precipitate crystalline inclusions---by metal alloying or atomic additions and then heat treatment
|
|
- Work-harden --- by processing and/or cold-working
|
|
- Decrease the grain size --- by processing and/or heat-treatment
|
|
|
|
### Make a Solid-Solution
|
|
|
|
Adding an alloying element, B, to the host, A, forms a solid-solution as the ions or atoms of B
|
|
dissolve in A.
|
|
|
|
The impurity particles of B are a different size from the particles of A, distorting the metal
|
|
lattice.
|
|
The larger the difference in radii of the particles, the bigger the distortion.
|
|
|
|
|
|
|
|
|
|
|
|
The particles of B tend to diffuse to dislocations and immobilise them.
|
|
This is why alloying increases the yield stress.
|
|
|
|
Impurity particles generate lattice strain in the structure too:
|
|
|
|
- Smaller particles introduce a compressive strain in the surrounding lattice
|
|
- Larger particles introduce a tensile strain in the surrounding lattice
|
|
|
|
|
|
|
|
### Precipitating Crystalline Inclusions
|
|
|
|
When adding an element, B, to a host, A, exceeds the solubility, the result is the formation of a
|
|
solid-solution with a fixed ratio of B to A, but also precipitated crystals of a different ratio of
|
|
B to A.
|
|
|
|
|
|
|
|
Crystalline inclusions are really difficult to shear, especially if they are small, numerous, and
|
|
have high Vickers' hardness.
|
|
This slows down dislocation movement, increasing yield stress.
|
|
|
|
### Work-Hardening and Cold Working
|
|
|
|
We can use room temperature deformation to increase the number of dislocations present in a metal.
|
|
As the % cold-work (%CW) is increased, the number of dislocations present also increases:
|
|
|
|
$$\% CW = \frac{A_0 - A_d}{A_0} \times 100\%$$
|
|
|
|
where $A_0$ is the initial cross sectional area and $A_d$ is the final cross sectional area.
|
|
|
|
A carefully prepared sample has a dislocation density, $\rho_d$ of around $10^3$ mm mm$3$,
|
|
whereas for a heavily deformed sample it is around $10^{10}$.
|
|
|
|
A high density of dislocations means they are more likely to get entangled with each other,
|
|
making it harder for dislocations to move.
|
|
Therefore as $\rho_d$ increases, yield stress does too.
|
|
|
|
### Decreasing the Grain Size
|
|
|
|
- Most metals are polycrystalline with many grains.
|
|
- Different grains will have a different crystal orientation.
|
|
- Grains impede dislocation motion
|
|
|
|
As you decrease grain size, you get more grain boundaries which basically creates more barriers
|
|
to prevent slip.
|
|
|
|
This is because a dislocation would have to change orientation across a grain boundary and "ionic
|
|
disorder in the grain boundary results in discontinuity of slip" (A.B Seddon University of
|
|
Nottingham 2020) (I think that's repeating it but it said it on the slideshow sooo...).
|
|
|
|
So for any given metal, the fine grained is harder and has greater yield stress than the coarse
|
|
grained version of it.
|
|
|
|
#### Hall Petch Equation
|
|
|
|
$$\sigma_{yield} = \sigma_0 + k_yd^{-0.5}$$
|
|
|
|
where $d$ is the grain size and $\sigma_0$ and $k_y$ are material constants.
|
|
|
|
Therefore a plot of $\sigma_{yield}$ against $d^{-0.5}$ would results in a straight line.
|
|
|
|
# Diffusion
|
|
|
|
Diffusion is atomic or ionic movement down a concentration gradient.
|
|
|
|
## Solid State Diffusion
|
|
|
|
|
|
|
|
Solid state diffusion is the stepwise migration (*march*) of atoms or ions through a lattice, from
|
|
site to site.
|
|
|
|
In order for this to happen, there must an adjacent vacant site.
|
|
The diffusion particle must also have sufficient thermal energy to 'jump' to the new site.
|
|
|
|
### Vacancy Diffusion (Diffusion of Metal Ions)
|
|
|
|
|
|
|
|
### Interstitial DIffusion (Diffusion of Small, Non-Metallic Particles)
|
|
|
|
|
|
|
|
## The Math(s) of Diffusion
|
|
|
|
Diffusion is time dependent.
|
|
|
|
For steady state diffusion, Fick's 1st Law holds:
|
|
|
|
$$J = -D \frac{\mathrm{d}C}{\mathrm{d}x}$$
|
|
|
|
where $J$ is the *flux*, $\frac{\mathrm{d}C}{\mathrm dx}$ is the concentration gradient, and $D$ is
|
|
the constant of proportionality known as the *diffusion coefficient*.
|
|
|
|
$D$ is constant for a particular metal at a particular temperature.
|
|
The *flux*is the number of atoms or ions moving per second through a cross sectional area.
|
|
|
|
### Things that Affect the Speed of Diffusion
|
|
|
|
- size of the diffusion species --- smaller species results in faster diffusion
|
|
- temperature --- more thermal energy allows more particles to have enough energy to make the 'jump'
|
|
- host lattice
|
|
|
|
- simple cubic --- 52% occupancy of ions
|
|
- body centered cubic --- 68% occupancy of ions
|
|
- face centered cubic --- 74% occupancy of ions
|
|
|
|
Diffusion is faster in a BCC host than in an FCC host for iron ions in an iron host and also for
|
|
carbon atoms diffusing into an iron host.
|
|
However this is not always the case.
|
|
|
|
### Influence of Temperature on Diffusion (Arrhenius Equation)
|
|
|
|
You can apply the Arrhenius equation for all thermally activated diffusion:
|
|
|
|
$$D = D_0 \exp{\left( - \frac{Q}{RT} \right)}$$
|
|
|
|
where $Q$ is the activation energy and $R$ is the ideal gas constant (8.31 J k$^{-1}$ mol$^{-1}$).
|
|
|
|
# Glossary
|
|
|
|
- liquidus - for a system of more than one component, the liquidus is the lowest temperature at
|
|
which the whole system is all in the liquid state.
|
|
- solidus - for a system of more than one component, the solidus is the highest temperature at which
|
|
the whole system is still in the solid state
|