## 4 Results
### Aqueous Saline Solutions  
#### 4.2.1 Presentation  
Given the previous analysis and practical remarks on the interpretation of DSC curves, it is now proposed to extend the purpose to binary mixtures that undergo their phase transition on a temperature range and not at a fixed temperature anymore. The first example we present concerns the case of saline solutions which is particular because the solid phases are pure. In this section, we treat the case where the solid phase is the ice, the cases where the solid phases would be the salt will lead to similar conclusions. The initial mass concentration of the solute (salt) \(x_{0}\) is defined by:  
\[x_{0}=\frac{m_{a}^{0}}{m_{w}^{0}+m_{a}^{0}} \tag{14}\]  
\(m_{a}^{0}\) being the initial mass of the solute and \(m_{w}^{0}\) being the initial mass of the solvent (water). In the goal to present general results independent of the nature of the solute, we will give same results considering the molar concentrations (for the ionic solutes it will be, in fact, osmolar concentrations) defined as:  
\[y_{0}=\frac{n_{a}^{0}}{n_{w}^{0}+n_{a}^{0}} \tag{15}\]  
\(n_{a}^{0}\) being the initial number of moles of the solute and \(n_{w}^{0}\) being the initial number of moles of the solvent (water). Figure 16: Schematic demonstration of the non-existence of any hysteresis phenomenon for a pure substance. (**a**) Initial thermograms for heating (red) and cooling (blue); (**b**) Symmetry over time; (**c**) Sign change for heat flux (endo vs. exothermic). For example, in Figure 17 it is schematized the phase diagram of a binary solution presenting an eutectic equilibrium. We also present, in this figure, two cases of the enthalpy following the investigated temperature range. When the melting begins at a temperature lower than the eutectic temperature \(T_{E}\), we have the function depicted in the left. The value of the enthalpy when the melting begins at a temperature higher than the eutectic temperature is given on the right. Above the eutectic temperature \(T_{E}\), the enthalpy corresponds to a progressive melting (equilibrium, at \(T\), between the pure ice and the solution whose mass concentration \(x(T)\) is given by the phase diagram (see Figure 17)). This progressive melting ends at \(T_{\text{liquidus}}\) which is the temperature of the equibrium of pure ice and the solution at the initial mass concentration \(x_{0}\). The phase diagram permits to determine the mass fraction of liquid water at the local temperature \(T\) (see Figure 17) defined as:  
\[\chi_{w}^{L}=\frac{m_{w}^{L}}{m_{w}^{0}+m_{a}^{0}} \tag{16}\]  
by the equation:  
\[\chi_{w}^{L}(T)=x_{0}\left(\frac{1}{x(T)}-1\right) \tag{17}\]  
and \(\chi_{w}^{L}(T_{\text{liquidus}})=1-x_{0}\). For the sake of clarity, in this section:  
* we reduce the temperature range to avoid the eutectic melting. We recall that the eutectic melting occurs at fixed temperature and all the results described for pure substances are valuable and we have already presented results concerning eutectic meltings [13; 14]. So the enthalpy function  
Figure 17: Phase diagram of an aqueous solution with an eutectic equilibrium and the specific enthalpy with (**left**) and without (**right**) the eutectic melting. corresponds only to the progressive melting (equilibrium, at \(T\), between the pure ice and the solution whose mass concentration is \(x(T)\)). This progressive melting ends at \(T_{\rm liquidus}\) which is the temperature of the equiprium of ice and the initial mass concentration \(x_{0}\) (see Figure 17). * we limit the range of concentrations \(x_{0}\) to the small values where the liquidus is a straight line. From a practical point of view, this does not alter the generality of the conclusions but only simplify the calculations and the analysis, the number of meaningful parameters being reduced. With these assumptions, the enthalpy function is:  
\[h(T)=\left\{