# Thermal Characterization of Phase Change Materials by Differential Scanning Calorimetry: A Review
## 3 Calorimetric Characterization
### Calorimetric Methods
\begin{table}
\begin{tabular}{p{12.3 pt} p{128.0 pt} p{128.0 pt} p{128.0 pt} p{128.0 pt}} \hline
**Test Type** & **Sample Size** & **Calibration** & **Advantages** & **Disadvantages** & **References** \\ \hline \multirow{4}{*}{Heat-flux DSC} & \multirow{4}{*}{(10–100 \(\mu\)L)} & All reference materials for melting point and enthalpy, and sapphire for heat capacity & Widely available, wide temperature range from sub-ambient to hundreds of degrees & Small sample size & Small sample size & \\  & & & & \\ \hline \multirow{4}{*}{T-history} & \multirow{4}{*}{(10–20 mL)} & \multirow{4}{*}{Water (Cp), Copper (Cp), Indium (T, Ah) [28]} & Self-built, can cover large samples & Time-consuming (one heating/cooling cycle can be done within a day) & Time-consuming (one heating/cooling cycle can be done within a day) & Time-consuming (20, 22, 28, 29) \\  & & & & & \\ \cline{1-1} \cline{3-4}  & & & & & \\ \cline{1-1} \cline{3-4}  & & & & & \\ \cline{1-1} \cline{3-4}  & & & & & \\ \hline \multirow{4}{*}{3-layer calorimeter (3 LC)} & \multirow{4}{*}{(100 mL)} & Only by the manufacturer with (water and hexadecane) & Uniquely built for large samples of PCM & Only available from a single company Temperature range (\(-\)20 to 100 \({}^{\circ}\)C) & Only available from a single company Temperature range (\(6\),\(26\)) \\ \cline{1-1} \cline{3-4}  & & & & & \\ \cline{1-1} \cline{3-4}  & & & & & \\ \cline{1-1} \cline{3-4}  & & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: Summary of the different calorimetric methods. Figure 3: hf-DSC: (**a**) DSC furnace including sample and reference crucibles; (**b**) schematic of crucibles and sensors inside a DSC chamber (personal data). As mentioned earlier, a calorimeter can supply heat into the sample by electrical or thermal resistance [17]. In heat-flux DSCs, heat is supplied to the sample through a thermal resistance (\(R_{th}\) in K\(\cdot\)W\({}^{-1}\)) and the differences between the ambient of the furnace and the sample \(\dot{Q}_{a-s}\), and the ambient of the furnace and the reference \(\dot{Q}_{a-r}\), are measured using Equations (3) and (4). Therefore, the difference between the heat flow into the sample and the reference is given by Equation (5). \[\dot{Q}_{a-s}=\frac{1}{R_{th}}(T_{a}-T_{s}) \tag{3}\]  
\[\dot{Q}_{a-r}=\frac{1}{R_{th}}(T_{a}-T_{r}) \tag{4}\]  
\[\dot{Q}=\dot{Q}_{a-s}-\dot{Q}_{a-r}=\frac{1}{R_{th}}(T_{r}-T_{s}) \tag{5}\]  
where the indexes \(a\), \(r\) and \(s\) stand for the ambient, reference and sample, respectively. It can be observed in Equation (5) that the heat flow difference \(\dot{Q}\) is not dependent on the ambient temperature [17]. Considering the isothermal condition for the sample and reference, one gets:  
\[\dot{Q}_{a-s}=\dot{Q}_{s}=C_{s}\frac{dT_{s}}{dt} \tag{6}\]  
\[\dot{Q}_{a-r}=\dot{Q}_{r}=C_{r}\frac{dT_{r}}{dt} \tag{7}\]  
where \(C\) is the mass heat capacity (J\(\cdot\)K\({}^{-1}\)). Assuming the same heating rate (\(\beta\) in \({}^{\circ}\)C\(\cdot\)min\({}^{-1}\)) in DSC's furnace (sample, reference and ambient), Equation (9) is obtained:  
\[\beta=\frac{dT_{s}}{dt}=\frac{dT_{r}}{dt}=\frac{dT_{a}}{dt} \tag{8}\]  
\[\dot{Q}_{a-s}-\dot{Q}_{a-r}=\frac{dT_{s}}{dt}C_{s}-\frac{dT_{r}}{dt}C_{r}=\beta (C_{s}-C_{r}) \tag{9}\]  
The proportionality of the temperature difference \(\Delta\)T between the sample and the reference to their heat capacities can be noticed by combining Equations (5) and (9):  
\[\beta(C_{s}-C_{r})=\frac{1}{R_{th}}(T_{r}-T_{s}) \tag{10}\]  
Figure 3:0  
The value of the thermal resistance (\(R_{th}\)) is determined by calibration with a reference material, such as indium, tin, etc [18]. In order to represent enthalpy, the first step is to determine the specific heat capacity using Equation (13). The heat flow difference is measured by the DSC and the sample mass and heating rate are known.