# Hansen Solubility Parameters (HSP)

The Hansen Solubility Parameters (HSP) were initially developed by Charles M. Hansen in 1967 as a tool for predicting if one material will dissolve in another and form a solution. Today, they are not only limited to predict solubility of chemical components but they are also used to optimize the stability of particle suspensions or the adhesion of polymers on surfaces. Therefore, the HSP were considered as a useful means to accelerate the development of formulations of inks, paints, adhesives, cosmetics or pharmaceutics.

The HSP are based on the idea that ‘like dissolves like’ or, in the case of solid surfaces to ‘like seeks like’. Each component, molecule or solid surface is described by three HSP parameters ($\delta$), generally measured in $MPa^{1/2}$. The three $\delta$ parameters are also called cohesion energy parameters:

- $\delta _d$ The energy from dispersion forces between molecules
- $\delta_p$ The energy from dipolar intermolecular forces between molecules
- $\delta _h$ The energy from hydrogen bonds between molecules.

These three parameters can be treated as coordinates for a point in three dimensions also known as the Hansen space. The nearer two molecules are in this three-dimensional space, the more likely they are to dissolve into each other. To determine if the parameters of two molecules [usually a solvent ($_2$) and a polymer ($_1$)] are within range, a value called interaction radius ($R_0$) is given to the substance being dissolved. This value determines the radius of the sphere in Hansen space and its center is the three Hansen parameters. To calculate the distance ($R_a$) between Hansen parameters in Hansen space the following formula is used:

$$R_a^2=4(\delta _{d2}-\delta _{d1})^2 + (\delta _{p2}-\delta _{p1})^2 + (\delta _{h2}-\delta _{h1})^2$$

The relative energy difference ($RED$) of the system is defined as follows:$$RED=\frac {R_a}{R_0}$$

If $RED<1$, the components are alike and will dissolve. If $RED=1$ the system will partially dissolve. If $RED>1$, the system will not dissolve.

## Hansen Solubility Parameters and IGC

The HSP are related to the interaction ability of molecules with other species. On the other side, IGC-ID is a method dedicated used to measure the interactions between a solute and a stationary phase (liquid or solid). Therefore, methods to determine the Flory-Huggins interaction parameter ($\chi_{1,2}^\infty$) and the HSP values of polymers from the IGC-ID measurements have been first developed by Lipson and Guillet.

The proposed method consists in placing equally a controlled amount of the polymer to characterize on an inert support, like Chromosorb^{®} or silanized glass beads. The measurements are generally carried out at temperature significantly above the $T_g$ of the polymer. They consist in determining the retention times ($t_N$) of three families of solvents: apolar, polar and those able of hydrogen bonding.

These retention times are then converted into specific retention volumes ($V_g^°$) as follows:

$$V_g^°=\frac {273.15}{T}.\frac {D_c}{m_s} .t_N$$

with $T$, the measurement temperature [$K$], $D_c$ the corrected carrier gas flow and $m_s$ the weight of deposited polymer.

Several relationships have been proposed in order to compute $\chi _{1,2}^\infty$ from the measured $V_g^°$ values. We use the one proposed by Adamska and Voelkel:$$\chi _{1,2}^\infty =\ln \left (\frac{273.1}{M_1.V_g^° P_1^0}.R\right )-\frac {P_1^0}{R.T} (B_{11}-V_1^0 )+\ln \left( \frac {\rho_1}{\rho _2}\right)- \left (1-\frac {V_1^0}{V_2^0} \right)$$

The above equation indicates that numerous of physicochemical parameters of both molecular probes and coating (or polymer) are needed. Some of them must be computed using other physicochemical characteristics. This is the case of B^{11} the second Virial coefficient of the probe.

The difficulty comes from the fact that some data are difficult to find. Sometimes, several sets of data exist in the literature and all of them are not suitable. To overcome these difficulties we choose to use the Software and database proposed by the professor Abbott: HSPiP.

This Software is able to compute the HSP starting from any experimental $V_g^°$ data.

Of course, in order to perform the above computations, the density ($\rho _2$) and the molar volume ($V_2^0$) of the polymer must be known.

The solubility parameter can also be calculated using the method proposed by Voelkel [2007] using the following equation:$$\frac {\delta_1^2}{RT}- \frac {\chi _H^\infty}{V_1^0}= \frac {2 \delta_2}{RT}.\delta _1- \frac {\delta _2^2}{RT}$$

with $\chi _{1,2}^∞=\chi _S^\infty +\chi_H^\infty$, $\delta _1$ the solubility parameter of the probe and $\delta _2$ the solubility parameter of the investigated substance. Assumptions are made about the value of $\chi _S^\infty$ in order to accede to $\chi _H^\infty$.

The tested solutes used in IGC-ID experiments can be divided into three groups, representing different intermolecular interactions: dispersive, polar and hydrogen bonding.

Plotting a linear relation for each group of tested solutes, according to the above equation, the components values ($\delta _{2d}$, $\delta _{2p}$ and $\delta _{2h}$) of the total solubility parameter $\delta _{2}$ can be calculated from the slope ($S_\chi$) of a straight line using the following relationships:

$$\delta _{2d}=\frac {S_{n-alkanes}}{2}RT$$$$\delta _{2p}=\frac {(S_{polars}-S_{n-alkanes})}{2}RT$$$$\delta _{2h}=\frac {(S_{h-bonds}-S_{n-alkanes})}{2}RT$$$$\delta ^2_2=\delta ^2_{2d} +\delta ^2_{2p} + \delta ^2_{2h}$$

Obviously, the great advantage of the use of IGC-ID is the quickness of the HSP value determination. Moreover, the required amounts of tested solvents and of the investigated material are very low.