Revisiting Hansen Solubility Parameters by Including Thermodynamics

Abstract The Hansen solubility parameter approach is revisited by implementing the thermodynamics of dissolution and mixing. Hansen's pragmatic approach has earned its spurs in predicting solvents for polymer solutions, but for molecular solutes improvements are needed. By going into the details of entropy and enthalpy, several corrections are suggested that make the methodology thermodynamically sound without losing its ease of use. The most important corrections include accounting for the solvent molecules’ size, the destruction of the solid's crystal structure, and the specificity of hydrogen‐bonding interactions, as well as opportunities to predict the solubility at extrapolated temperatures. Testing the original and the improved methods on a large industrial dataset including solvent blends, fit qualities improved from 0.89 to 0.97 and the percentage of correct predictions rose from 54 % to 78 %. Full Matlab scripts are included in the Supporting Information, allowing readers to implement these improvements on their own datasets.


Introduction
Solvents are extremelyi mportanti nm any industrial sectors. As chemists, we tend to think mainly about chemical reactions and active ingredients. But in real-world terms, the solvent and formulation of ac hemical product, be it an ink, ap aint, an oil derivative, or ah erbicide, makes the bulk of the product. Often, it is also crucial for the function of the product:t he formulationi sw hat makes ap esticide stay longero nt he leaves after rain, determines the distribution and drying of paints and inks, or controls the applicabilityo fh ealth andc osmetics products. Hence, knowing the solubility (or lack thereof) of chemicals is of utmost importance.
Moreover,when selecting as olvent, usually not only itsability to dissolve the active ingredient matters, but many other constraints apply as well (e.g. viscosity,v olatility,s ustainability, and safety of the formulation). Therefore, to efficientlys earch for satisfying formulations, ap redictive tool for solubility is indispensable. [1,2] The paint and coatings industry,f or example, has been using solubility parameters for many decades. [3] These are simple parameters for solvents and solutes based on the "like dissolves like" concept:w hen as olute and as olvent make similar interactions with each other as with their own kinds, there is no or little enthalpy loss upon mixing. The first solubility parameters were suggested by Hildebrand, who used as ingle parameter, d,b ased on the cohesive energy per volume [Eq. (1)]: [4] d ¼ where E cohesion = DH vap ÀRT.
As long as the interactions are purely dispersive,f or example, hydrocarbons versusf luorinated compounds, Hildebrand's methodi sr easonably accurate.H owever, for more general cases, it was Hansenw ho realized in the 1960s that the cohesive energy and the corresponding solubility parameters should be split into the three fundamental chemical interactions:d ispersion (D), polar interactions (P), and hydrogen bonding( H) [Eqs.
Hansen's pragmatic approach has earned its spurs in predicting solvents for polymer solutions, but for molecular solutes improvements are needed. By going into the details of entropy and enthalpy,s everalc orrections are suggested that make the methodology thermodynamically sound without losing itse ase of use. The most important corrections include accounting for the solventm olecules' size, the destruction of the solid's crys-tal structure, and the specificity of hydrogen-bonding interactions, as well as opportunities to predict the solubility at extrapolated temperatures.T esting the originaland the improved methods on al arge industrial dataset including solventb lends, fit qualities improved from 0.89 to 0.97 and the percentage of correct predictionsr ose from 54 %t o7 8%.F ull Matlab scripts are included in the Supporting Information,a llowing readers to implement these improvements on their own datasets.
Albeit indirectly,H ansenm easured the contribution of these three interaction types to the cohesive energy for many solvents. For solutes, the solubility parameters are determined by measuring the solubility at ag iven concentrationi naset of solvents. To do this, one first plots the solvent parameter values in at hree-dimensional space.T hen, as phere is fitted for the solute corresponding to the measured data (see Figure 1).
The radius of this sphere gives af ourth parameter that is specific for this solute and depends on the temperature and the concentration. Now,t he solubility in other solvents at this temperaturea nd concentration can be predicted by checking whether the parameters of the new solventf all inside or outside the sphere. Conveniently,f or solvent blends the parameters can be calculated by as imple linear interpolation based on the volume/volumer atio of the blend. When materials do not actually dissolve,s welling experiments [6,8] or inverse gas chromatography [9,10] may also be performed to find the Hansenparameters.
Hansen's methodology was first appliedi nt he paint and polymeri ndustries and was found to be satisfactory.M ore recently,H ansenp arameters have also been used in other sectors, to find solvents fora ll types of (small) molecules, including drugs, [11] cosmetics, [12] and oligomers, [13] as well as for predicting gel formation. [14][15][16] However,t he results are not as good when compared with those for polymers. Therea re two reasonsf or this:t he first is that drugs,c osmetics, and the like typicallyh ave more varied functional groups. The second reason is that the originalH ansenp arameters do not include thermodynamic considerations. This is acceptable for polymers (where the thermodynamics cancel out) but not for small molecule solutes.
Some rigorous thermodynamic derivations were published by Colemane tal. proving the correctness of Hildebrand's methodf or nonpolarp olymers and extending aH ildebrand/ Hanseni ntermediate (including polar interactions, but excluding hydrogen-bondingi nteractions) to nanotubes and nanosheets. [17] However,t hermodynamic corrections fors mall mole-cule effects and ac orrect handling of hydrogen-bonding interactions are still missing.
Here, we revisit the Hansens olubility parameter method for small molecules by adding thermodynamics to it. We introduce severalf undamental corrections that improvet he quality of the predictions. Our goal is to keep the conceptual simplicity of Hansen's method. Therefore, we refrain from including quantities that are difficult to measure, such as heats of melting or heat capacities. The new model is tested on al arge industrial dataset,a nd shows as ignificant improvement over the originalHansenmethod.

Theory
We present five improvements to Hansen's method,f ollowing from the actual thermodynamics of dissolution and mixing. When ac ompound dissolves, molecules leave the crystal and mix into the solvent. This gives ap ositivee ntropye ffect, but usually costs some enthalpy.T his enthalpy loss is relatedt o the distances in the Hansenp lots. The key issue here is that the amount of entropyg ained by mixing determines how much enthalpy can be lost while maintaining an egative DG. The distance that gives DG = 0d efines the radius of the Hansens phere. As the entropy effect depends on the concentration,t he temperature, and the size of the molecules, this should all be included in the methodology.O ur improvements are based on ab etter description of both the entropy and the enthalpy terms.
Note that the term "solubility spheres" is used;y et these spheres refer only to the parameter space (see Figure 1) and not to physical radii of molecules or other species.
Where x is the mole fractiono ft he solute and R is the ideal gas constant.Ast his is per mole, solvents with smallmolecules give ah ighere ntropyo fm ixing per liter of solvent( and are indeed knowna sb etter solvents). In the original Hansen method, however,a ll solvents are considered equallyg ood, and solubility only depends on the parameter distance from the solute.
Here, we introduce radii (in the parameter space)f or the solvents as well. These would be inversely correlated to the solvents' molarv olume. Figure 2s hows the concept of this improvement using the sum of the radii of the solute and the solventa sc utoff. In practice, we proposeu sing the inverse sum of these radii [Eq. (5)]: For each solute, asphereisf itted according to its solubility in these solvents.T he black dots denote the solvents that dissolvethe solute.
The rationale behind choosingt he inverse sum is thati t makes the smallervalue more important:ifthe solute is notoriously difficult to dissolve, it has as mall parameter radius and the effective radius should be small as well;i ft he solute is for instanceasolventi tself (with al arge parameter radius), the effective radius should be more equally weighted between the solute'sand the solvent's radii.

#2 Correcting for Concentration
Usually,H ansen spheres are fitted to data foro ne concentration. However,a se stablished above,t he entropy dependso n the mole fraction,a nd this may differ nonetheless as ar esult of the different molarv olumes of each solvent. Hence, we must correct for combining data at different mole fractions into one fit.
The radius of the Hansen sphere is defined by the ratio between entropya nd enthalpy of mixing [Eq. (6)]: where b is the so-called interaction parameter,a nd a is the activity coefficientc aused by non-ideal mixing (a = 1f or regular mixtures). If we now define the Hansen radius for ac ertain benchmark concentration, for example, x = 0.5 [Eq. (7)],w ec an calculate the radius at any other concentration by correcting for the change in the ratio between DS and DH [Eqs. (4) and (6)]: An additional advantage of this correction is that we may now combine data for different concentrations.

#3 Using the Squared Distance
As derived by Colemane tal. [17] and already mentioned by Hansen, [19] the enthalpy of mixingr elates to the square of the parameter distance:the interaction parameter, b,i s[ Eq. (8) where the subscripts Aa nd Br epresent solute and solvent parameters, respectively.N evertheless,H ansenu ses the unsquaredd istance to optimize the fit. In principle, this does not matter as squaring the radius does not affect as phere's shape. It only influences the weight of points that are missed by the fit. However,n ow that we have introduced solvents pheres, a different r eff appliesf or every solvent. Thus, now it does matter whether r eff or r eff 2 is used, so we stress that Equation (8) shouldb eu sed and not its square root.

#4 Splitting Donor and Acceptor Parameters
For hydrogen bonding, the reasoning" like dissolves like" is imprecise:h ydrogen bonds form between donors and acceptors, so to dissolve donors one needs acceptors, and vice versa. Therefore, the d H parametern eeds to be split into a d HD (donor) and a d HA (acceptor) parameter.N ow,t he cohesive energy contribution of hydrogen bonds (or any other Lewis acid-base interactions)i sd efined as [Eq. (9)]: Ar ecent version of the HSP software (v4.0, 2013) already attempts to include this effect, giving moderate improvements. There, the definition of d HD and d HA was inspired by the work of Abraham, including the reasoningt hat Equation (9) does not allow for an on-zero d H combined with az ero d HD . [20,6] Hence,t hey defined the relationd ifferently: d H 2 = d HD 2 + d HA 2 . But if d H is not equal to zero, then d HD (or d HA )c annot be zero either:i ft he experimental cohesive energy shows that there are non-zero Lewis acid-base interactions in the pure substance, then the molecules must have both Lewis acid and base character (even if chemical intuition suggests they do not).
Following from Equation (9), the hydrogen-bonding contributiont othe interaction parameter, b,becomes [Eq. (10)]: In this way,w er ecognize the advantageous mixingo fd onor speciesw ith acceptor species, in which case b can even become positive. However,s plit donor and acceptorp arame- ters for the solvents are very hard to obtain experimentally,a s they are coupled. Following Hunter,though, they can be calculated from the electrostatic potential in electronic structure calculations. [21] Hunter compiledatable with donora nd acceptor strengthso fm any functional groups. But as we do not want to include specific knowledge of the number of functional groups in each solvento rsolute molecule, we need volumebased parameters. Therefore, for each solvent, we took the average of the values of all the functional groups in the molecule and scaled this to fit with the known value for d H (see the Supporting Information ford etails). This averageda pproach is approximate, yet it works reasonably well (cf. the work of Beerbower et al.,w ho estimated Lewis acid and base components based on spectroscopic measurements [22] ).

#5 IncludingE nthalpy and Entropy of Melting
Many solutes are solid at room temperature. This meanst hat upon dissolution, there are not only enthalpy and entropye ffects of the mixing itself, but also of the destruction of the crystal structure.T hus, the enthalpy loss of meltinga nd mixing needs to be overcome by the entropyg ain of melting and mixing.Note that these are the virtual enthalpy and entropy of meltinga tt he mixingt emperature, which may differ from those at the meltingp oint owing to differences in heat capacity. [23] Putting this into equations, we should realize that the enthalpy and entropy of meltinga re defined per mole of solute, whereas the mixingq uantities are defined per total number of molecules [Eq. (11)]: On the limit of solubilityb oths ides are equal, andE quation (12) applies: We now replace ðDH melt;T À TDS melt;T Þ with ac onstant, c melt , and, because the Hansenr adius is related to b instead of DH mix ,w ed ivide by x(1Àx)[ see Eq. (6)]: The first parto ft his equation is not influenced by the solute melting, and can be replaced by Equation (5) [or Eq. (7)] to give Equation (14): We can thus correct for the melting of the solute by subtractingasimple (temperature-dependent) constant, c melt divided by (1Àx). The constant c melt is the free energy lost by breaking down the crystal structure of the solid solute. However,i ts value is unknown (because of the aforementioned heat capacity effects)a nd needs to be optimized while fitting the data. Moreover, owing to the same effects, c melt depends unpredictably on T. In principle, though, c melt should alwaysb ep ositive, because DH melt is larger than TDS melt at any T below the melting point.

The Effect of Temperature
As we now have better descriptions of entropya nd enthalpy, we can also predict their dependence on temperature. For moderate temperatured ifferences, we can assume the solubility parameters, d D , d P ,a nd d H ,t ob econstant. This assumption is not valid for large temperature differences, though: [7] in the liquid state,a th igher temperatures the molecules will rotate more, giving less optimal polar interactions, and hydrogen bonds will break more often. As ar esult, the cohesive energy resultingf rom polar (d P )a nd hydrogen-bonding (d H )i nteractions will decrease.T his effect can be estimated by calculating the Boltzmann weighted averages over the molecular orientations of polar compounds and over the number of broken hydrogen bonds. [24] Note, however,t hat the same effect occurs in all the liquid states:pure solvent, molten solute, and solution.The major relative effect is to be expected for the solids olute, as in the solid phase rotational freedom and broken hydrogen bonds are less frequent. However, this differenceg oes into the c melt parameter and not into the solubility parameters, d. Hence, we believe that the relative effect (besides af ormal shift for all solvents and solutes) of temperature on the solubility parameters, should be small, and will depend on the actual combination of strongly and weakly binding functional groups in as pecific solute-solvent pair.
Within the approximation of regular mixing, neither the entropy nor the enthalpy of mixing dependo nt he temperature. Therefore, the only effect is the dependence of the Gibbs free energy itself: DG = DHÀTDS. As the solubility radius is essentially the (corrected) distance in the parameter space for which DH = TDS (i.e. DG = 0), all solubility radii (r solute and r solvent ) depend linearly on T [Eq. (15)]: That said, there is one complication:t he c melt constant introduced in Equation (13) In practice, the heatc apacity varies and tends to increase strongly close to the melting point. As ar esult,f or some solutes, when far from the melting point, it may be better to consider c melt to be constant with T,w hich follows from substitut-  (17): As the behavior of the heat capacity is very compound-specific, choosing between Equations (16) and (17) is difficult. Therefore, we pragmatically suggest using the average of these two equations [Eq. (18)]: For Equations (16) and (18), the meltingp oint of the solute is needed.H owever,a si ti su nclear which equation is the best to use in the first place, an estimate of the melting point suffices.

InterpolationofParameters for Blends
An important advantage of the Hansen methodology is that it can easily predict solubilitiesi ns olvent blends as well. It is common practice to estimate the parameters fors uch blends by linear interpolation of the parameters of the constituting pure solvents. Nowt hat we have improved the method from pragmatically effective to thermodynamically sound, we can check the correctness of such interpolations. Also, we need mixing rules for the new parameters that we introduced: r solvent , d HD ,a nd d HA .W riting down the full derivations (see the Supporting Information) shows that the correct mixingr ule is indeed approximately al inear interpolation based on the volume/volume ratio of the solvents in the blend. In particular, we find that the same mixing rule appliesfor all parameters.

Concerningt he Correction Factor for d D
In the original Hansens olubility methodology,a ll d D values are multipliedb ya na dditional empirical factor of 2,s imply because this improves the agreement with experimental data. [5][6][7] However,u ntil today no physical explanation wasf ound for this correction factor. [17] Incidentally,a bove we explained that temperature-dependent Boltzmann weighing of molecular orientations will lower the effective d P and d H values. This might be the physical explanation of Hansen's correction factor.H owever,o ne should realize that this effect dependsh ighly on the strength of specific interactions and on the actual temperature. Aconstantcorrectionf actor of 2i ssurely unphysical.

Methods
We tested our improved methodology on an industrial dataset of 15 solutes of very different types, including herbicides, organometallic complexes, electrolytes, polymers, and other compounds. The solutes' solubility parameters were determined by measuring their solubility at fixed concentrations in as tandard set of 48 solvents (with parameters ranging from 12.9-21.0 for d D ,0 -26.2 for d P ,a nd 0-42.3 for d H ). Some of the solutes were tested at multiple concentrations, resulting in at otal of 21 fit sets. Experiments in other ran-domly chosen solvents (172 data points) and solvent blends (284 data points) were used as ap rediction set. All experiments were performed at room temperature by al iquid handling robot using 1.00 mL solvent per vial, and the degree of dissolution (complete or incomplete) was determined visually after mixing for 24 h.
For four out of the 15 solutes, we also ran experiments at lower temperatures, ranging from 4 8Cd own to À10 8C. These samples were first mixed at room temperature. After mixing for 24 h, they were cooled for 24 h; then ac rystalline seed of the solute was added followed by another 24 ho fc ooling, before the degree of dissolution was determined. In total, four fit sets and ap rediction set of 125 data points were collected at lowered temperatures.
Before being able to fit the solute parameters, ap riori values for the solvents are needed for the new parameters, r solvent , d HD ,a nd d HA .T he radius r solvent was assumed to be inversely proportional to the molar volume, V M ,a nd was calibrated on solvent-solvent mixing data, resulting in Equation (19): When comparing partly improved methodologies, r solvent was calibrated for each choice of settings. The parameters d HD and d HA were estimated as described in the Supporting Information and were scaled to fit with the known values for d H .F urthermore, for calculating the mole fraction, x,o fasolute its molar mass needs to be known. For polymers, any large number can be used, as for large masses the actual value makes little difference;w eu sed 10 10 gmol À1 (we checked that using 10 4 gmol À1 gives practically the same results, but if the molar mass is known it is clearly best to use the true value). Note that c melt only applies to solutes, so no apriori values are needed.
The fitting was performed by steepest-descent optimization of d D , d P ,a nd d H ,c ombined with stepwise adjustment of r, c melt ,a nd the d HD /d HA ratio. However,o wing to the Boolean nature of the data (complete vs. incomplete dissolution) the optimized function has discontinuous derivatives, making robust optimization difficult. To overcome this, we performed optimizations from ac oncentric grid of 129 starting points around the average of all good solvents and selected the solution with the best fit, usually leading to better fits than with the HSP software. In the case of perfect fits, the average was taken of all perfect solutions found. For fair comparison, the same algorithm was used both for testing the original Hansen method and our improved methodology.
Note that the fitting procedures for Hansen spheres are themselves as ubject of discussion. [25][26][27] Moreover,t he usual definition of the optimum is problematic:when two data points are misfitted at opposite sides of the Hansen sphere, any solution between these two points gets the same score. Improvements of the fitting procedure itself should also focus on this linear definition of the optimum, but this is outside the scope of this work. More details of our fitting procedure as well as the script for implementing it are included in the Supporting Information.
The quality of the fit, q F ,w as calculated as proposed by Hansen: [19] for data points at the wrong side of the sphere, the error is calculated by calculating the distance of this point to the edge of the sphere (or actually by calculating the difference between the interaction parameter b and the value for b that would place the data point exactly on the edge of the sphere Here, n is the total number of data points in the fit (not only the ill-fitted points). Ap erfect fit gets av alue of 1; non-perfect fits get lower values. Note that the values are corrected for the absolute scale of the radii, where r solvent is any solvent's radius calibrated for the original method and r' solvent is the equivalent radius calibrated for ag iven set of improvements. In fact, for the fully improved method the calibrated scale is very similar to that of the original method.

Results
To establish the power of Hansen's originalm ethod and our improved method, we assessed both the ability to fit known data as well as the quality of predictions for unseen solvents based on these fits. Also, the relative importance of the various theoreticali mprovements wasa ssessed by testing subsets, where the numbers refer to the numberingi nt he theory section. Note that improvements #3 and #5 are effective only when improvements #1 or #2 are used. Similarly,i mprovement #4 only makes sense in combination with #3. Figures 3a nd 4 show the ability to fit known data, expressedb yt he percentage of data points that could not be fitted at the correct side of the sphere, and by the fit quality, q F .A lthough the improvement in the number of missed points is modest, q F improves impressively,w hichm eanst hat points that could not be fitted, still are approached much closer. As high fit scores may alwaysb ec aused by overfitting, [28] the mosti mportant results are the predictionso ft he solubility in unseen solvents and solvent blends ( Figure 5). These are strongly improved as well. Interestingly,i mprovements #1 and #5 apparently are the most important.T he combination of only these two improvements already leads to a q F of 0.96 and 77 %c orrect predictions. With respect to the d D correction factor,w en otet hat, although the percentage of missed data pointsd uring fitting becomes better when the factor is included, the quality of the predictions actually gets worse. Interestingly,w ef ound that the predictions are even more improved for solvent blends than for pure solvents. However,a st his refers to differentd ata points, such numbersc annot be compared. Therefore, we did not analyze the difference between pure solvents and blends.
Overall, our completei mproved method( improvements #1 + 2 + 3 + 4 + 5) leads to an improvement of the q F going from 0.89 for Hansen's originalm ethod to 0.97, which is almostf our times closer to perfectf itting (q F = 1). Moreover, the percentage of correctp redictionsi mproved from 54 %t o 78 %. Realizing that random predictions will also be correct for 50 %o ft he points,t he improvementisl arge.

Predicting the Solubility at Extrapolated Temperatures
The dependences of the solubility parameters, including r and c melt ,o nt he temperature, allow predictions to be made at extrapolated temperatures. This was tested by predicting solubilities at lowered temperatures based on experimental tests at room temperature. For comparison, the solubility in the standard test set of 48 solvents was also measured at the lowered temperature for predictions withoute xtrapolation. Note that the percentages of correct predictions are lower than in Figure 5. This is because only af ew of the standard solvents dissolve the solutes at lowered temperature, resulting in inaccurate fitted spheres. Although the statistics for these tests are rather limited, Figure 6shows that the extrapolated predictions are at least as good as the predictions at constant temperature. In fact, the extrapolatedp redictionsa re better. The latter reflects the fact that at higher temperatures more solvents can dissolve ag iven solute, which gives more information for fitting and thus am ore precise fit. Note that this improvement for the extrapolatedp redictionsi ss tatistically relevant, whereas the difference between the original method and the improvedm ethodi sn ot statistical relevant for this (small) low T dataset.

Discussion
Methods for predictings olubility are typically very generalized to make predictionsf or new solutes and unseen solvents. On the other hand, enough accuracyi sn eeded to be of practical value. We show that by considering the thermodynamics of solubility,s imple generalized modelsc an be improved significantly without sacrificing the general applicability.T hat said, generalized methods are never exact. For instance, most solubility modelsassume regular mixing, but reality is full of effects that break ideality that cannot be accounted for in at heoretical framework. For this reason, we stuck to the Hansen concept of fitting, as this should capture at least parto ft hese non-idealities without incorporating them explicitly.
In recent years, several other modelsh ave been used for predicting solubility fully theoretically.E xamples are:U ni-Quac, [29] COSMO-RS, [30] and even ac ombination of COSMO and Hansen. [31] However, COSMO modelso nly aim at improving the description of the enthalpy terms. Therefore, these methods could also be improvedf urther by including entropya nd melting corrections, similarly to what we have shown here. We do note that fully theoretical methods will be haunted by non-idealities to al arger extent than ap rocedure of fitting to real experimental data such as the Hansenm ethodology.H ansen parameters can also be estimated theoretically,a ss tudied extensively by Panayiotou et al. [32,33] They even already included estimates for acidic and basic components. [34] Indeed, parameters calculated with Panayiotou's methodc an in principle be used in our method when experimental solventp arameters are not available.
For our data set with very different types of solutes, we improvedt he quality of the predictions enormously compared with Hansen's original method. Therefore, in retrospect, one might be surprised that Hansen's methodw orkedi nt he first place for the paint and polymer industries. The simple reason is that for polymers the solute solubility radii are very small, utterly overshadowing the importance of the solvent radii, and thus corrections #1, #2, and #5 have essentially no effect. For small solute molecules, however, these corrections are highly important,a ss hown by our results. Nevertheless, there is still room for improvements. Av ery practical point is that Hansen's definition of the best fit is ill defined as it is based on linear errors, as already discussed in the Methods section. As ar esult, the fitting procedure lacks somer obustness and our predictions may not yet be optimal.
Moreover,o ur implementation of donor and acceptorp arameters is still imperfect. Many of the solutes ands olvents in our data set have multiple functional groups and their parameters are now averaged into one donor ando ne acceptor parameter.O nt op of that, large molecules with one strongd onating group or small molecules with aw eaker donating group have the same parameter values, although their interactions with acceptorm olecules may differ depending on the properties of the acceptor.E ven though fitting of averaged donor and acceptor parameters was shown to be effective, [35] more correct would be to work out all hydrogen-bonding pairs from both molecules.B ut implementing that into as olubility parameter approachisn ot straightforward.
Another approach to solubility predictions is the work of Ruelle and Huyskens andc o-workers. [36,37] They introduced a very interesting approacht oc alculate entropye ffects that are influenced by preferential interactions such as hydrogen bonds. Implementing their treatment of entropy into our im-provedH ansen methodology would probablyg ive af urther improvement.H owever,tod ot hat, we would again have to let go of the averagedh andling of multiple groups, so implementation into our approach is again not straightforward. Figure 6. Predictiono fsolubility at low temperatures (4 8CtoÀ10 8C). "low T from low T"means that training sets and prediction sets are obtained at the same temperature." Low T from room T"means that the samedata set is predicted from training sets obtained at room temperature, using the opportunity to extrapolate the temperature in the improved method. Af inal opportunity to further improve our methodology would be to add at emperature-dependent Boltzmann weighing term to the polar and hydrogen-bonding interactions,a s already discussed in the Theory section. Nevertheless, with the improvements proposed in the current work we already improve significantly upon the originalH ansen method, as witnessed by the results. It should be noted, though, that these improvements are based on the thermodynamics of dissolution. Other applications of Hansen parameters, such as swelling, permeability,o rg el formation have different thermodynamics, so different corrections may apply.

Conclusions
The philosophy of Hansen solubility parameters is correct in using parameters for the three fundamental chemical interactions. However, the originalm ethodology lacks thermodynamics, making it less suitable for non-polymerics olutes.H ere, we have made Hansen's method thermodynamically sound by adding simple corrections based on the entropya nd enthalpy of mixing andm elting.
Our improvedm ethodology was tested on an industriald ataset of 15 solutes of very different types, for which the solubility was predicted in unseen solvents and solvent blends. Compared with Hansen's originalm ethod, the percentage of correct predictions for this dataset improved impressively from 54 %t o7 8%.T he most important corrections leading to this large improvement were found to be the introduction of solvent solubility radii and ac orrection for the "melting" of solid solutes.A lso, split donor and acceptor parameters were applied, but the success of this was somewhat hampered by the necessity to averageo vers everal functional groups per molecule.
Additionally,w eh ave shown that with the improved methodology predictions can be made at extrapolated temperatures. Alli na ll, we have made al arge step in improvingH ansen's solubility parameter approach for solubility predictions of polymers as well as non-polymeric solutes in unseen solvents and solventblends.