add notes on thermal expansion, thermal conductivity

mechanical/mmme1029_materials.md

@@ -631,3 +631,143 @@ 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>