title: MMME1029 // Materials

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Lecture 1 (2021-10-04)

1A Reading Notes

High Temperature Materials (and Theoretical Thermodynamic Efficiency)

Types of Stainless Steel

Self Quiz 1

  1. What is made of billion year old carbon + water + sprinkling of stardust?

    Me

  2. What are the main classifications of materials?

    Metals, glass and ceramics, plastics, elastomers, polymers, composites, and semiconductors

  3. [There are] Few Iron Age artefacts left. Why?

    They rusted away

  4. What is maens by ‘the micro-structure of a material’?

    The very small scale structure of a material which can have strong influence on its physical properties like toughness and ductility and corrosion resistance

  5. What is a ‘micrograph’ of a material?

    A picture taken through a microscope

  6. What microscope is used to investage the microstructure of a material down to a 1 micron scale resolution?

    Optical Microscope

  7. What microscope is used [to investigate] the microstructure of a material down to a 100 nm scale resolution?

    Scanning Electron Microscope

  8. What length scales did you see in the first slide set?

    1 mm, 0.5 mm, 1.5 m

  9. What material properties were mentioned in the first slide set?

    Hardness, brittleness, melting point, corrosion, density, thermal insulation

Self Quiz 2

  1. What is the effect of lowering the temperature of rubber?

    Makes it more brittle, much less elastic and flexible

  2. What material properties were mentioned in the second slide set?

    Young’s modulus, specific heat, coefficient of thermal expansion

Lecture 2

Properties of the Classes

Metals

Ceramics and Glasses

Polymers

Composites

Terms

Organic vs Inorganic Materials

Crystalline vs Non-Crystalline Materials

Material Properties

Density

ρ=mv\rho = \frac m v

Melting Points

Corrosion

Self Quiz

Consolidation Questions 1

    1. Metal
    2. Titanium
    1. Polymer
    2. Polyester
    1. Ceramic
    2. Alumino-silicate
    1. Composite
    2. GFRP or CFRP
    1. Metal
    2. Aluiminium alloy
    1. Metal
    2. Aluiminium alloy
    1. Polymer
    2. Acrylic
    1. Ceramimc
    2. Glass
    1. Composite
    2. Concrete

Consolidation Questions 2

C B

Polymers

Introduction to Polymers

There are 3 types of polymers:

Polymers are relatively new materials, lightweight, durable, flammable, and degraded by UV light. They are made of long carbon-carbon chains.

Stress-Strain Curve of Polymers

Industrially Important Polymers

The worldwide production of polymers in 2019 was 368×106368\times10^6 tonnes and the majority is from just 5 polymers:

All of these materials are low cost.

Thermoplastics

The simplest polymer is poly(ethene):

When 2 polymer chains get close together, Van der Waals (vdw) forces keep them together. vdw forces are very weak, much weaker than the covalent bonds inside the polymer.

Stress Strain Curve

Crystalline and Amorphous/Glassy Solids (Heating and Cooling)

Amorphous Thermoplastics

Crystalline Polymers

Semi-Crystalline

Specific Volume vs Temperature

Path ABCD

Path ABEF

Relative Molar Mass and Degree of Polymerisation

where

Making Polymers

There are two ways to make polymers:

Elastic Deformaion

Elastic deformation is deformation where the material will return to original shape after the applied stresses are removed.

Elastic deformation is the first type of deformation that happens when stresses are applied to a material and is represented by the straight line at the beginning of a stress-strain curve.

Modulus of Resillience (ErE_r)

This is the area under the elastic portion of a stress-strain graph of a material.

Plastic Deformation

Toughness (Absorbing Energy Through Plastic Deformation)

Ductility measures how much something deforms plastically before fracture, but just because a material is ductile does not make it tough.

The key to high material toughness is a good combination of large ultimate fracture stress and large ductility.

Charpy Impact Test

Measures material toughness by determining the amount of energy absorbed during fracture.

It works by essentially dropping a hammer into a sample whose dimensions are standardized (usually either by BSI or ISO) and measuring how high the hammer goes up on the other side, after it breaks the material

The height of the hammer after impact will tell you how much enery is left in it, and therefore how much has been aborbed by the now broken sample.

Under a microscope, more ductile fractures appear fibrous or dull, whereas less ductile surfaces have granular or shiny surface texture.A

The charpy test has a couple issues:

The setup of a charpy impact test

  1. Sample is made to standardized dimensions, with a notch
  2. Sample is placed on support
  3. A very heavy hammer pendulum of mass mm is dropped from rest at h0h_0 to swing about a pivot, reaching EkmaxE_{kmax} vertically below the pivot.

    1. If no sample is in place then the hammer will swing back up on the other side to a height of hhh_h where theoretically hh=h0h_h = h_0

    2. With a sample placed vertically below, some of the EkE_k is transferred to the sample to bend and (usually) break the sample.

      If breaks the sample, it will swing up to the other side, where its max height, hfh_f can be used to calculate how much energy was used to break the sample:

      E=mg(hh−hf)E = mg(h_h-h_f)

      Where gg is acceleration due to gravity.

Ductility

Ductility is the plastic deformation a material withstands before fracture.

Griffith Surface Flaws

These flaws vary in size and shape. They limit the ability of any material, brittle or ductile, to withstand tensile stresses as they concentrate the tensile forces applied to a smaller area.

The stress at the tip of the flaw:

σactual=2σar\sigma_{actual} = 2\sigma\sqrt{\frac a r}

For deep (aa is large) or thin (rr is small) the stress is magnified and, if it exceeds the UFS in a brittle material, the flaw will grow into a crack, resulting in the brittle material fracturing.

However in a ductile material, the tip of the flaw can heal, reducing aa and increasing rr. This is due to the chemical structure of ductile materials like metals.

Stress Intensity Factor

Stress Intesity Factor, KK:

K=fσπaK = f\sigma\sqrt{\pi a}

where:

Fracture Toughness

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]

The value of KK that causes the notch to grow and cause fractures. This is value is known as the fracture toughness, KcK_c.

At low thicknesses fracture toughness depends on thickness but as thickness increases, KcK_c decreases to the constant value, the plane strain fracture toughness, K1cK_{1c}.

Composites

Composites are made of two or more materials, which when combined together, at up to a milimetre scale, have superior properties to their parent materials.

Composites tend to be 2-phase: a dispersed phase in a matrix. The disepersed phase tends to be fibres (large aspect ratio) or particles (low aspect ratio) which are embedded in a matrix, which are often resins.

Composite properites are affected by the dispersed phase geometry:

Rule of Mixtures

EcE_c lies between the arithmetic mean (upper limit):

VmEm+EpVpV_mE_m + E_pV_p

and the geometric mean (lower limit):

VmEmEpVpVmEm+EpVp\frac{V_mE_mE_pV_p}{V_mE_m + E_pV_p}

Where EcE_c, EmE_m, EpE_p are the Young’s moduluses of the composite, matrix, and particles, respectively, and VmV_m and VpV_p are the volume of the matrix and particles, respectively.

Particle Reinforced Composites

Applications of Composites

Tungsten Carbide Cobalt for Cutting Tools

The Tungest Carbide (WC) particle are a truly brittle ceramic. They are very hard but the brittleness means they are easy to break.

The solution is to hold small WC particles in a ducitle metal matrix. In this case it is Cobalt (Co).

This way, crack in one WC particle does not necessarily mean other particles are broken, meaning the cutting tool overall still works.

Another advantage of this composite is that WC is not very thermally conductive and has a high melting point, which allows it to work well the environment it’s in.

Resin Bonded Alumina for Sanding Disks

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.

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:

Influence of the Fibres

Depends on:

Stress Strain Graph of a Fibre Reinforced Composite

Under uniaxial, longitudinal loading in tension

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

Thermal Properties of Materials

Specific Heat Capacity

How much heat energy is required to raise the temperature of a body by one unit:

Cp=ΔEmΔT C_p = \frac{\Delta E}{m\Delta T}

where cc is specific heat capacity.

It is measured at a constant pressure, usually 1.013×1051.013\times 10^5 Pa.

Molar Heat Capacity

Cpm=ΔEnΔTC_pm = \frac{\Delta E}{n\Delta T}

What is a mole?

The mole (symbol: mol) is the base unit of amount of substance in the International System of Units (SI). It is defined as exactly 6.02214076×10236.02214076\times 10^{23} elementary entities (“particles”)

~ Wikipedia: Mole (unit)

How Much Does a mol of Something weigh?

A mol of an element weighs its relative atomic mass (ArA_r) but in grams. For example, Carbon-12 has an ArA_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.

Thermal Expansion

Origin of Thermal Expansion

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 Graph

Morse potential is the energy well between 2 bonded atoms. The graph is asymmetric due to the repulsion experienced by atoms as they apporach.

Linear Coefficient of Thermal Expansion

αL=ΔLL0ΔT\alpha_L = \frac{\Delta L}{L_0 \Delta T}

where LL is the sample length.

Example 1

A 1 m long bar of aluminium metal cools in the solid state from 660 C to 25 C.
Calculate the length of the bar after it cools down, given αL=25×10−6\alpha_L = 25\times10^{-6} K−1^{-1}.

l0=1ΔT=Tf−T0=25−660=−635αL=lf−l0l0ΔTαLl0ΔT=lf−l0lf=αLl0ΔT+l0=0.984\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*}

Linear Thermal Expansion and Isotropism

Since isotropic solids have the same properties in all directions, you can say that for an isotropic solid:

αV=3αL=ΔVV0ΔT\alpha_V = 3\alpha_L = \frac{\Delta V}{V_0 \Delta T}

Reasons to Care About Thermal Expansion

Thermal Conductivity

Thermal conductivity is the rate at which heat power is transferred through a material.

QA=kΔTΔx\frac{Q}{A} = k \frac{\Delta T}{\Delta x}

where QQ is heat power, AA is area of the surface, ΔTΔx\frac{\Delta T}{\Delta x} is the temperature gradient, and kk is the thermal conductivity constant.

Origin of Thermal Conductivity

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.

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

Cubic Unit Cells

Packing Factor

packing factor=ions per unit cell×VionVcell\text{packing factor} = \frac{\text{ions per unit cell} \times V_{ion}}{V_{cell}}

Theoretical Density

theoretical density=ions per unit cell×mionVcell\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:

FeBCC↔FeFCC\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, λ\lambda, is roughly equal to the distance, dd, between atom/ion layers. This allows x-rays to probe for dd via Bragg’s Equation:

Requirements for the x-rays:

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, 2dsinθ2d\sin\theta is any multiple, nn, of λ\lambda. Or:

nλ=2dsinθn\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:

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.

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.

Acid etched surface of a polycrystalline metal

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

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.

Substitutional addition replaces ions in the host
Interstitial addition adds particles between the ions in the host

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:

How Ni content in Cu affects Yield and Ultimate Tensile Stress

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=A0−AdA0×100%\% CW = \frac{A_0 - A_d}{A_0} \times 100\%

where A0A_0 is the initial cross sectional area and AdA_d is the final cross sectional area.

A carefully prepared sample has a dislocation density, ρd\rho_d of around 10310^3 mm mm33, whereas for a heavily deformed sample it is around 101010^{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 ρd\rho_d increases, yield stress does too.

Decreasing the Grain Size

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

σyield=σ0+kyd−0.5\sigma_{yield} = \sigma_0 + k_yd^{-0.5}

where dd is the grain size and σ0\sigma_0 and kyk_y are material constants.

Therefore a plot of σyield\sigma_{yield} against d−0.5d^{-0.5} would results in a straight line.

Heat Treatment of Metals

These processes are to change a material’s mechanical properties, not change its shape.

Phase Diagrams

Here is an example of a two component phase diagram with a familiar system:

The component in this case are sugar and water, but not syrup.

A phase is a chemically and physically distinct species as we can have a change in phase that goes from solid to solid.

The solubility limit is the maximum concentration for which only a solution occurs. In the case of this system, thee limit increases with temperature.

Here is a generic phase diagram for a generic A-B system:

Annealing

Annealing is a process by which a component is heated to remove the effects of cold work.

These are diffusional processes and only occur at high temperatures. The driver for diffusion is the removal of high energy defects from the system.

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=−DdCdxJ = -D \frac{\mathrm{d}C}{\mathrm{d}x}

where JJ is the flux, dCdx\frac{\mathrm{d}C}{\mathrm dx} is the concentration gradient, and DD is the constant of proportionality known as the diffusion coefficient.

DD is constant for a particular metal at a particular temperature. The fluxis the number of atoms or ions moving per second through a cross sectional area.

Things that Affect the Speed of Diffusion

Influence of Temperature on Diffusion (Arrhenius Equation)

You can apply the Arrhenius equation for all thermally activated diffusion:

D=D0exp(−QRT)D = D_0 \exp{\left( - \frac{Q}{RT} \right)}

where DD is the diffusion coefficient, D0D_0 is the frequency factor, QQ is the activation energy, RR is the ideal gas constant (8.31 J k−1^{-1} mol−1^{-1}).

You can find the diffusion distance, xx, with the following equation:

xDtx ~ \sqrt{Dt}

Materials in Sustainable Transport

Is the car emissions reduction target significant?

Overall CO2 emissions in 2016 is 466 Megatonnes.

Does a reduction from 130 g / km to 95 g / km (a 35 g/km reduction) make a significant difference?

There are 33 million registered cars in the uk.

If they average around 8000 miles each (~13000 km) per year that’s a ~15 Megatonne reduction, or about 3% of the annual C02 emmissions, a significant reduction.

Materials in Cars

Material Substitution

Case Study — 2012 Honda Accord

Choosing a Material

Glossary

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