Hansen Solubility Parameters

For over 50 years, Hansen Solubility Parameters have provided guidance to a large range of solubility issues: finding good solvents, identifying suitable plasticisers, understanding polymer interactions and so forth. For each material three parameters are required: δD, the Dispersion (van der Waals) component, δP, the Polar component and δH, the Hydrogen bonding component. Tables of these for common solvents can be found on the Hansen Solubility site and on my Practical Solubility site.

Measurement of the HSP of solids is relatively straightforward. Against a test range of solvents spanning HSP space the solubility of the solid (polymer, nanoparticle, pharmaceutical...) is scored as either good or bad and from the set of solubilities, the HSP can be estimated as a sphere containing all the good solvents and excluding all the bad ones.

Although this technique can be used on semi-solids such as oligomers or the sorts of excipients commonly used in pharma and cosmetics, the measurements are less successful because the solubility sphere is too large. For these materials IGC promises to be a superior methodology.

A set of IGC measurements can be carried out using a neutral packing coated with a thin layer of the test material with a set of probes carefully-chosen to span HSP space. This gives their Vg values (specific retention volumes) which in turn are normalised values calculated from retention times. These values in turn can be transformed into a χ parameter (actually, the Flory-Huggins χ₁₂ parameter) that describes how alike the probe and the test molecule are. The transformation, unfortunately, is rather complex:

`χ = ln(R.(273+T)/(M_1.Vg.P_1))-P_1/(RT).(B_(11)-V_1)+ln(ρ_1/ρ_2)-(1-V_1/V_2)`

M₁, P₁, ρ₁ and V₁ are the MWt, vapour pressure, density and Molar Volume of the probe, B₁₁ is the second Virial coefficient of the probe and ρ₂ and V₂ are the density and Molar Volume of the test material.

Fortunately the key values for all the probes are calculable from data incorporated within the app. You just need to know the Vg values for the probes, T_m, the temperature of the measurement and ρ₂ and V₂. A representitive set of data are provided as default when you first start the app.

So for each probe we have a measured χ. But χ depends on the three HSP of the probe and the test material, in particular on the HSP Distance D which is defined as:

`D^2 = 4(δD_1-δD_2)^2+(δP_1-δP_2)^2+(δH_1-δH_2)^2`

From D we obtain χ as:

`χ = (D^2.V_2)/(4RT)`

So extracting the HSP from the data "simply" requires comparing the measured versus calculated χ parameters and varying the HSP of the test material till the fit is optimal. In the app this is done via trial and error but it clearly can be done via a fitting algorithm.

The theory is excellent. In practice we see lots of issues from people trying to extract HSP data via IGC:

• They extract δD via an alkane plot then extract δP and δH via a different plot. This is unsatisfactory in two ways. First, δD can vary from 14 to 20 but alkanes are all around 15 so you are not statistically challenging δD space.
• Second, too small a range is used. Typically 5-6 alkanes are used, which add very little new information, then, say, 4 alcohols, which again don't add much information beyond just one, then an ether and a ketone. To be statistically valid the probes should span a broad HSP range. So use just one or two alkanes one or two alcohols, leaving time for an ether, a ketone, a chlorosolvent, an aromatic, a cyclic carbonate, acetonitrile ... .
• The assumption is that the probe is in thermodynamic equilibrium with the test material. This can be false for two reasons. First, if the support material is not completely covered, the effective (average of support+test) system will have a very different HSP. Second, if the probes pass through too quickly, those that should interact strongly (higher Vg) will come out too quickly so their χ will be too large. The best HSP datasets have carefully checked that both issues are under control.

The data shown are an excellent set from Tian and Munk and represent Polycaprolactone at 70 °C. The best fit (you can aim to maximise the R² fitting value where 1 is perfect) is somewhere around 19, 5, 6

The IGC apps are based on the inputs kindly provided by Dr Eric Brendlé of Adscientis who are specialists in IGC measurements.