mmme1029 Notes on the crystallisation of materials

mechanical/mmme1028_statics.md

@@ -500,7 +500,7 @@ E &= \frac \sigma \epsilon = 2513836.686 = 2.5\times10^6 \text{ Pa}
 
 ## Poisson's Ratio
 
-For most materiajs, their cross sectionts change when they are stretched or compressed.
+For most materials, their cross sectionts change when they are stretched or compressed.
 This is to keep their volume constant.
 
 $$ \epsilon_x = \frac {\Delta L}{L_0} $$

mechanical/mmme1029_materials.md

@@ -631,3 +631,274 @@ bonds between the surface of the fibre and matrix.
   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 Graph](./images/vimscrot-2021-12-21T19:51:58,667328620+00:00.png)
+
+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):
+
+  ![](./images/vimscrot-2021-12-21T21:28:34,863875469+00:00.png)
+
+  Lattice Parameter = 2r
+
+- Face centred cubic (FCC)
+
+  ![](./images/vimscrot-2021-12-21T21:44:21,618384089+00:00.png)
+
+- Body centred cubic (BCC)
+
+  ![](./images/vimscrot-2021-12-21T21:44:40,816535537+00:00.png)
+
+### 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
+
+![](./images/vimscrot-2021-12-21T22:33:35,491930818+00:00.png)
+
+### 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*
+
+![](./images/vimscrot-2021-12-21T22:40:37,978916142+00:00.png)
+
+
+## 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:
+
+![](./images/vimscrot-2021-12-21T22:44:15,147729727+00:00.png)
+
+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.

mechanical/mmme1048_fluid_mechanics.md

@@ -466,3 +466,41 @@ M_{OO} &= F_py_p = \int_{area}\! \rho gh^2 \,\mathrm{d}A \\
 \\
 y_p = \frac{M_{OO}}{F_p}
 \end{align*}
+
+## Buoyancy
+
+### Archimedes Principle
+
+> The resultant upwards force (buoyancy force) on a body wholly or partially immersed in a fluid is
+> equal to the weight of the displaced fluid.
+
+When an object is in equilibrium the forces acting on it balance.
+For a floating object, the upwards force equals the weight:
+
+$$mg = \rho Vg$$
+
+Where $\rho$ is the density of the fluid, and $V$ is the volume of displaced fluid.
+
+### Immersed Bodies
+
+As pressure increases with depth, the fluid exerts a resultant upward force on a body.
+There is no horizontal component of the buoyancy force because the vertiscal projection of the body
+is the same in both directions.
+
+### Rise, Sink, or Float?
+
+- $F_B = W$ \rightarrow equilirbrium (floating)
+- $F_B > W$ \rightarrow body rises
+- $F_B < W$ \rightarrow body sinks
+
+### Centre of Buoyancy
+
+Buoyancy force acts through the centre of gravity of the volume of fluid displaced.
+This is known as the centre of buoyancy.
+The centre of buoyancy does not in general correspond to the centre of gravity of the body.
+
+If the fluid density is constant the centre of gravity of the displaced fluid is at the centroid of
+the immersed volume.
+
+![](./images/vimscrot-2021-12-21T15:08:22,285753421+00:00.png)
+