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Title: Revisiting Hansen solubility parameters by includingthermodynamics

Authors: Manuel J Louwerse, Ana G. Maldonado, Simon Rousseau,Chloe Moreau-Masselon, Bernard Roux, and GadiRothenberg

This manuscript has been accepted after peer review and appears as anAccepted Article online prior to editing, proofing, and formal publicationof the final Version of Record (VoR). This work is currently citable byusing the Digital Object Identifier (DOI) given below. The VoR will bepublished online in Early View as soon as possible and may be differentto this Accepted Article as a result of editing. Readers should obtainthe VoR from the journal website shown below when it is publishedto ensure accuracy of information. The authors are responsible for thecontent of this Accepted Article.

To be cited as: ChemPhysChem 10.1002/cphc.201700408

Link to VoR: http://dx.doi.org/10.1002/cphc.201700408

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Improved by thermodynamics: Hansen solubility parametersworkwellforpredictingthesolubilityofpolymers,butlesswellforthat of small molecules. By considering the thermodynamics ofdissolutionandmixing,weimprovethemethodologyaswellasthequalityofthepredictions.

RevisitingHansensolubilityparametersbyincludingthermodynamicsManuelJ.Louwerse1,2,AnaMaldonado1,3,SimonRousseau4,ChloeMoreau-Masselon4,BernardRoux3,andGadiRothenberg1,*1Van‘tHoffInstituteforMolecularSciences,UniversityofAmsterdam,Postbus94720,1090GSAmsterdam,TheNetherlands2InorganicChemistryandCatalysis,DebyeInstituteforNanomaterialsScience,UtrechtUniversity,Universteitsweg99,3584CG,Utrecht,TheNetherlands3Solvay,Research&InnovationCenter,178Av.DrAlbertSchweitzer,F-33608Pessac,France4Solvay,Research&InnovationCenterParis/Novecare,52ruedelahaieCoq,F-93308Aubervilliers,FranceAbstractTheHansensolubilityparameterapproachisrevisitedbyimplementingthethermodynamicsofdissolutionandmixing.Hansen’spragmaticapproachhasearneditsspursinpredictingsolventsfor polymer solutions, but formolecular solutes improvements are needed. By going into thedetails of entropy and enthalpy, several corrections are suggested thatmake themethodologythermodynamicallysoundwithoutlosingitseaseofuse.Themostimportantcorrectionsincludeaccountingforthesolventmolecules’size,thedestructionofthesolid’scrystalstructure,andthespecificityofhydrogenbondinginteractions,aswellasopportunitiestopredictthesolubilityatextrapolatedtemperatures.Testingtheoriginalandtheimprovedmethodsonalargeindustrialdatasetincludingsolventblends,fitqualitiesimprovedfrom0.89to0.97andthepercentageofcorrect predictions rose from 54% to 78%. FullMatlab scripts are inlcuded in the supportinginformation,allowingreaderstoimplementtheseimprovementsontheirowndatasets.KEYWORDS: Solubility parameters, thermodynamics, entropy, temperature, donor-acceptorinteractionsINTRODUCTIONSolvents are extremely important in many industrial sectors. As chemists, we tend to thinkmainlyaboutchemicalreactionsandactiveingredients.Butinreal-worldterms,thesolventandformulationofachemicalproduct,beitanink,apaint,anoilderivative,oraherbicide,makesthebulk of the product. Often, it is also crucial for the function of the product: the formulation iswhat makes a pesticide stay longer on the leaves after rain, determines the distribution anddryingofpaintsandinks,orcontrolstheapplicabilityofhealthandcosmeticsproducts.Hence,knowingthesolubility(orlackof)ofchemicalsisofutmostimportance.Moreover,whenselectingasolvent,usuallynotonly itsabilitytodissolvetheactive ingredientmatters, butmany other constraints apply as well (e.g. viscosity, volatility, sustainability, andsafetyoftheformulation).Therefore,toefficientlysearchforsatisfyingformulations,apredictivetoolforsolubilityisindispensable.[1,2]The paint and coatings industry, for example, has been using solubility parameters for manydecades.[3]Thesearesimpleparametersforsolventsandsolutesbasedonthe“likedissolveslike”

10.1002/cphc.201700408ChemPhysChem

This article is protected by copyright. All rights reserved.

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concept:whenasoluteandasolventmakesimilarinteractionswitheachotheraswiththeirownkinds, there is no or little enthalpy loss upon mixing. The first solubility parameters weresuggested by Hildebrand, who used a single parameter, δ, based on the cohesive energy pervolume:[4]

! = !cohesion!mol

(1)

WhereEcohesion=ΔHvap−RT.As long as the interactions are purely dispersive, e.g. hydrocarbons versus fluorinatedcompounds,Hildebrand’smethodisreasonablyaccurate.However,formoregeneralcases,itwasHansen who realized in the 1960s that the cohesive energy and the corresponding solubilityparameters should be split into the three fundamental chemical interactions: dispersion (D),polarinteractions(P),andhydrogenbonding(H):[5-7]

!cohesion = !D + !! + !! (2)

!! = !D! + !P! + !H! (3)Albeit indirectly, Hansen measured the contribution of these three interaction types to thecohesive energy for many solvents. For solutes, the solubility parameters are determined bymeasuringthesolubilityatagivenconcentrationinasetofsolvents.Todothis,onefirstplotsthesolvent parameter values in a three-dimensional space. Then, a sphere is fitted for the solutecorresponding to the measured data (see Figure 1). The radius of this sphere gives a fourthparameterthatisspecificforthissoluteanddependsonthetemperatureandtheconcentration.Now,thesolubilityinothersolventsatthistemperatureandconcentrationcanbepredictedbychecking whether the parameters of the new solvent fall inside or outside the sphere.Conveniently, for solvent blends the parameters can be calculated by a simple linearinterpolation based on the volume/volume ratio of the blend.Whenmaterials do not actuallydissolve,alsoswellingexperiments[6,8]orinversegaschromatography[9,10]maybeperformedtofindtheHansenparameters.

Figure 1. Three-parameter plot showing the solubility sphere in the original Hansen method:Solventsareplottedaccordingtotheirsolubilityparameters,δD,δP,andδH.Foreachsolute,asphereisfittedaccordingtoitssolubilityinthesesolvents.Theblackdotsdenotethesolventsthatdissolvethesolute.Hansen’s methodology was first applied in the paint and polymer industries and foundsatisfactory.Morerecently,Hansenparametersarealsousedinothersectors,tofindsolventsforalltypesof(small)molecules, includingdrugs,[11] cosmetics,[12] andoligomers,[13] aswellasforpredictinggel formation.[14-16]However,theresultsarenotasgoodwhencomparedwiththosefor polymers. There are two reasons for this: The first is that drugs, cosmetics and the liketypically have more varied functional groups. The second reason is that the original Hansenparametersexcludethermodynamicconsiderations.This isacceptable forpolymers(wherethethermodynamicscancelout)butnotforsmallmoleculesolutes.



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Some rigorous thermodynamic derivations were published by Coleman et al. proving thecorrectnessofHildebrand’smethodfornonpolarpolymersandextendingaHildebrand/Hansenintermediate (including polar interactions, but excluding hydrogen bonding interactions) tonanotubes and nanosheets.[17] However, thermodynamic corrections for smallmolecule effectsandacorrecthandlingofhydrogenbondinginteractionsarestillmissing.Here we revisit the Hansen solubility parameter method for small molecules by addingthermodynamicstoit.Weintroduceseveralfundamentalcorrectionsthatimprovethequalityofthepredictions.OurgoalistokeeptheconceptualsimplicityofHansen’smethod.Therefore,werefrain from includingquantities thataredifficult tomeasure, suchasheatsofmeltingorheatcapacities. The new model is tested on a large industrial dataset, and shows a significantimprovementovertheoriginalHansenmethod.THEORYWepresentfiveimprovementstoHansen’smethod,followingfromtheactualthermodynamicsofdissolutionandmixing.Whenacompounddissolves,moleculesleavethecrystalandmixintothesolvent.Thisgivesapositiveentropyeffect,butusuallycostssomeenthalpy.Thisenthalpylossisrelated to thedistances in theHansenplots. Thekey issuehere is that the amount of entropygainedbymixingdetermineshowmuchenthalpycanbe lostwhilekeepinganegative∆G.Thedistance that gives ∆G=0 defines the radius of the Hansen sphere. Since the entropy effectdependsontheconcentration,thetemperature,andthesizeofthemolecules,thisshouldallbeincluded in themethodology.Our improvements arebasedonabetterdescriptionofboth theentropyandtheenthalpyterms.Note that the term ‘solubility spheres’ is used; yet these spheres refer only to the parameterspace(seeFigure1above)andnottophysicalradiiofmoleculesorotherspecies.Improvementsrelatedtoentropy(1)IntroducingsolventradiiAssumingregularmixing,theentropyofmixingpermoleoftotalmaterial(soluteplussolvent)is:[18]

∆!mix = −! ! ln ! + 1 − ! ln 1 − ! (4)Wherex isthemolefractionofthesoluteandR istheidealgasconstant.Sincethisispermole,solventswithsmallmoleculesgiveahigherentropyofmixingperlitersolvent(andareindeedknownasbettersolvents). IntheoriginalHansenmethod,however,allsolventsareconsideredequallygood,andsolubilityonlydependsontheparameterdistancefromthesolute.Hereweintroduceradii(inparameterspace)forthesolventsaswell.Thesewouldbeinverselycorrelatedtothesolvents’molarvolume.Figure2showstheconceptofthisimprovementusingthe sum of the radii of the solute and the solvent as cutoff. In practice,we propose using theinversesumoftheseradii:

!eff = !!

!solute! !!solvent

(5)

The rationale behind choosing for the inverse sum is that it makes the smaller value moreimportant:ifthesoluteisnotoriouslydifficulttodissolve,ithasasmallparameterradiusandtheeffectiveradiusshouldbesmallaswell;ifthesoluteisforinstanceasolventitself(withalargeparameter radius), the effective radius should bemore equallyweighted between the solute’sandthesolvent’sradii.



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Figure 2. Schematic showing thephilosophybehind including solvent radii: thealloweddistancenow also depends on the solvent radius. Note that using the inverse summation (and other non-linearcorrectionsdiscussedbelow)precludesanyfurthergraphicaldataanalysis.(2)CorrectingforconcentrationsUsually,Hansenspheresarefittedtodataforoneconcentration.However,asestablishedabove,theentropydependsonthemolefraction,andthismaydiffernonethelessduetodifferentmolarvolumesofeachsolvent.Hence,wemustcorrect forcombiningdataatdifferentmolefractionsintoonefit.TheradiusoftheHansensphereisdefinedbytheratiobetweenentropyandenthalpyofmixing,where: ∆!mix = −!"# 1 − ! (6)βistheso-calledinteractionparameter,andαtheactivitycoefficientcausedbynon-idealmixing(α=1 for regular mixtures). If we now define the Hansen radius for a certain benchmarkconcentration,e.g.x=0.5,wecancalculatetheradiusatanyotherconcentrationbycorrectingforthechangeintheratiobetweenΔSandΔH(eqs.4and6):

!eff(!) =∆!mix ! ∆!mix !

∆!mix !=0.5 ∆!mix !=!.!!eff(!=0.5) =

! !"!! !!! !" !!!! !!!

! !"(!.!) !eff(!=0.5) (7)

An additional advantage of this correction is that we may now combine data for differentconcentrations.Improvementsrelatedtoenthalpy(3)UsingthesquareddistanceAs derived by Coleman et al.[17] and alreadymentioned by Hansen,[19] the enthalpy of mixingrelatestothesquareoftheparameterdistance:theinteractionparameter,β,is:

! = − !! (!D,A − !D,B)! + (!P,A − !P,B)! + (!H,A − !H,B)! (8)

where the subscripts A and B represent solute and solvent parameters, respectively.Nevertheless,Hansenusestheunsquareddistancetooptimizethefit. Inprinciplethisdoesnotmattersincesquaringtheradiusdoesnotaffectasphere’sshape.Itonlyinfluencestheweightofpoints that are missed by the fit. However, now that we have introduced solvent spheres, adifferentreffappliesforeverysolvent.Thus,nowitdoesmatterwhetherrefforreff2isused,sowestressthateq.8shouldbeusedandnotitssquareroot.(4)SplittingdonorandacceptorparametersFor hydrogen bonding, the reasoning ‘like dissolves like’ is imprecise: Hydrogen bonds formbetween donors and acceptors, so to dissolve donors one needs acceptors, and vice versa.Therefore,theδHparameterneedstobesplitintoaδHD(donor)andaδHA(acceptor)parameter.Now the cohesive energy contribution of hydrogen bonds (or any other Lewis acid-baseinteractions)isdefinedas:



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!cohesion,H = !HD!HA (9)ArecentversionoftheHSPsoftware(v4.0,2013)alreadyattemptstoincludethiseffect,givingmoderate improvements. There, the definition of δHD and δHA was inspired by the work ofAbraham, including the reasoning thateq.9doesnotallow foranon-zeroδHcombinedwithazeroδHD.[20,6]Hence,theydefinedtherelationdifferently:δH2=δHD2+δHA2.ButifδHisnotequaltozero, then δHD (or δHA) cannot be zero either: If the experimental cohesive energy shows thattherearenon-zeroLewisacid-baseinteractionsinthepuresubstance,thenthemoleculesmusthavebothLewisacidandbasecharacter(evenifchemicalintuitionsuggeststheydonot).Followingoneq.9,thehydrogenbondingcontributiontotheinteractionparameter,β,becomes: !H = !

! (!HD,A!HA,B + !HD,B!HA,A − !HD,A!HA,A − !HD,B!HA,B) (10)In this waywe recognize the advantageousmixing of donor species with acceptor species, inwhichcaseβ canevenbecomepositive.However, splitdonorandacceptorparameters for thesolvents are very hard to obtain experimentally, since they are coupled. Following Hunter,though, they can be calculated from the electrostatic potential in electronic structurecalculations.[21]Hunter compileda tablewithdonorandacceptor strengthsofmany functionalgroups.Butsincewedon’twanttoincludespecificknowledgeofthenumberoffunctionalgroupsin each solvent or solute molecule, we need volume-based parameters. Therefore, for eachsolvent,wetooktheaverageofthevaluesofallthefunctionalgroupsinthemoleculeandscaledthis to fitwith the knownvalue forδH (see supporting information for details). This averaged approach is approximate, yet it works reasonably well (cf. the work of Beerbower et al., whoestimatedLewisacidandbasecomponentsbasedonspectroscopicmeasurements[22]).Otherimprovements(5)IncludingenthalpyandentropyofmeltingManysolutesaresolidatroomtemperature.Thismeansthatupondissolution,therearenotonlyenthalpy and entropy effects of the mixing itself, but also of the destruction of the crystalstructure.Thus, the enthalpy lossofmelting andmixingneeds tobeovercomeby the entropygainofmeltingandmixing.Notethatthesearethevirtualenthalpyandentropyofmeltingatthemixingtemperature,whichmaydifferfromthoseatthemeltingpointduetodifferencesinheatcapacity.[23] Putting this into equations, we should realize that the enthalpy and entropy ofmeltingaredefinedpermoleofsolute,whilethemixingquantitiesaredefinedpertotalnumberofmolecules: ! ∆!mix + !∆!melt,! ≥ ∆!mix + !∆!melt,! (11)Onthelimitofsolubilitybothsidesareequal,and: ∆!mix = !∆!mix − !(∆!melt,! − !∆!melt,!) (12)We now replace(∆!melt,! − !∆!melt,!)with a constant, cmelt, and, because the Hansen radius isrelatedtoβinsteadof∆!mix,wedividebyx(1−x)(seeeq.6): !eff = !∆!mix

!(!!!) −!melt!!! (13)

Thefirstpartofthisequationisnotinfluencedbythesolutemelting,andcanbereplacedbyeq.5(oreq.7): !eff = 'concentration_correction'

!!solute

! !!solvent

− !melt!!! (14)

We can thus correct for the melting of the solute by subtracting a simple (temperaturedependent)constant,cmeltdividedby(1−x).theconstantcmeltisthefreeenergylostbybreakingdown the crystal structure of the solid solute. However, its value is unknown (because of theaforementionedheatcapacityeffects)andneedstobeoptimizedwhilefittingthedata.Moreover,



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duetothesameeffects,cmeltdependsunpredictablyonT.Inprinciple,though,cmeltshouldalwaysbepositive,because∆!meltislargerthan!∆!meltatanyTbelowthemeltingpoint.TheeffectoftemperatureSince we now have better descriptions of entropy and enthalpy, we can also predict theirdependence on temperature. For moderate temperature differences, we can assume thesolubility parameters, δD, δP, and δH, to be constant. This assumption is not valid for largetemperaturedifferences,though:[7]Intheliquidstate,athighertemperaturesthemoleculeswillrotatemore,givinglessoptimalpolarinteractions,andhydrogenbondswillbreakmoreoften.Asa result, the cohesive energy due to polar (δP) and hydrogen bonding (δH) interactions willdecrease.ThiseffectcanbeestimatedbycalculatingtheBoltzmannweightedaveragesoverthemolecularorientationsofpolarcompoundsandoverthenumberofbrokenhydrogenbonds.[24]Note,however,thatthesameeffectoccursinalltheliquidstates:puresolvent,moltensolute,andsolution.Themajorrelativeeffectistobeexpectedforthesolidsolute,sinceinthesolidphaserotationalfreedomandbrokenhydrogenbondsarelessfrequent.However,thisdifferencegoesinto the cmelt parameter and not into the solubility parameters, δ. Hence, we believe that therelativeeffect(besidesaformalshiftforallsolventsandsolutes)oftemperatureonthesolubilityparameters,shouldbesmall,andwilldependontheactualcombinationofstronglyandweaklybindingfunctionalgroupsinaspecificsolute-solventpair.Within the approximation of regular mixing, neither the entropy nor the enthalpy of mixingdependonthetemperature.Therefore,theonlyeffectisthedependenceoftheGibbsfreeenergyitself:ΔG=ΔH−TΔS.Sincethesolubilityradiusisessentiallythe(corrected)distanceinparameterspaceforwhichΔH=TΔS(i.e.ΔG=0),allsolubilityradii(rsoluteandrsolvent)dependlinearlyonT: ! ! = !

!! !! (15)

Thatsaid,thereisonecomplication:thecmeltconstantintroducedineq.13alsodependsonthetemperature. If the heat capacity of the solid would be constant with varying T, cmelt woulddependlinearlyonT,butrelativetothemeltingpoint,TM: !melt ! = !M!!

!M!!!!melt! (16)

Inpractice,theheatcapacityvariesandtendstoincreasestronglyclosetothemeltingpoint.Asaresult,forsomesolutes,whenfarfromthemeltingpoint,itmaybebettertoconsidercmelttobeconstantwithT,whichfollowsfromsubstitutingTM=∞intoeq.16: !melt ! = !melt! (17)Asthebehavioroftheheatcapacityisverycompound-specific,choosingbetweeneqs.16and17isdifficult.Therefore,wepragmaticallysuggestusingtheaverageofthesetwoequations: !melt ! = !

!!M!!!M!!!

+ 1 !melt! (18)Foreqs.16and18,themeltingpointofthesoluteisneeded.However,sinceitisunclearwhichequationisthebesttouseinthefirstplace,anestimateofthemeltingpointsuffices.InterpolationofparametersforblendsAn important advantage of theHansenmethodology is that it can easily predict solubilities insolventblendsaswell.Itiscommonpracticetoestimatetheparametersforsuchblendsbylinearinterpolation of the parameters of the constituting pure solvents. Now that we improved themethodfrompragmaticallyeffectivetothermodynamicallysound,wecancheckthecorrectnessof such interpolation. Also,we needmixing rules for the newparameters thatwe introduced:rsolvent,δHD, andδHA.Writingdown the fullderivations (see supporting information) shows thatthe correct mixing rule is indeed approximately a linear interpolation based on thevolume/volumeratioof thesolvents in theblend.Especiallywe findthat thesamemixingruleappliesforallparameters.



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ConcerningthecorrectionfactorforδDIn the original Hansen solubility methodology, all δD values are multiplied with an additionalempirical factor of 2, simply because this improves the agreementwith experimental data.[5-7]However,untiltodaynophysicalexplanationwasfoundforthiscorrectionfactor.[17]Incidentally,aboveweexplainedthattemperature-dependentBoltzmannweighingofmolecularorientationswill lower the effective δP and δH values. This might be the physical explanation of Hansen’scorrectionfactor.However,oneshouldrealizethatthiseffectdependshighlyonthestrengthofspecific interactions andon the actual temperature.A constant correction factor of 2 is surelyunphysical.METHODSWe tested our improvedmethodology on an industrial dataset of 15 solutes of very differenttypes, including herbicides, organometallic complexes, electrolytes, polymers, and othercompounds.Thesolutes’solubilityparametersweredeterminedbymeasuringtheirsolubilityatfixedconcentrations ina standardsetof48solvents (withparameters ranging from12.9–21.0for δD, 0–26.2 for δP, and 0–42.3 for δH). Some of the solutes were tested at multipleconcentrations,resultinginatotalof21fitsets.Experimentsinotherrandomlychosensolvents(172 data points) and solvent blends (284 data points) were used as prediction set. Allexperiments were performed at room temperature by a liquid handling robot using 1.00 mLsolventpervial,andthedegreeofdissolution(completeorincomplete)wasdeterminedvisuallyaftermixingfor24h.For fouroutof the15 soluteswealso ranexperiments at lower temperatures, ranging from4downto–10°C.Thesesampleswerefirstmixedatroomtemperature.Aftermixingfor24h,theywerecooledfor24h;thenacrystallineseedofthesolutewasaddedfollowedbyanother24hofcooling,beforethedegreeofdissolutionwasdetermined.Intotal4fitsetsandapredictionsetof125datapointswerecollectedatloweredtemperatures.Beforebeingabletofitthesoluteparameters,apriorivaluesforthesolventsareneededforthenewparameters,rsolvent,δHD,andδHA.Theradiusrsolventwasassumedtobeinverselyproportionaltothemolarvolume,VM,andwascalibratedonsolvent–solventmixingdata,resultingin:

!solvent(!=0.5) ≈ !"## J mL/mol!M

(19)When comparing partly improved methodologies, rsolvent was calibrated for each choice ofsettings.TheparametersδHDandδHAwereestimatedasdescribedinthesupportinginformationand were scaled to fit with the known values for δH. Furthermore, for calculating the molefraction,x,ofasoluteitsmolarmassneedstobeknown.Forpolymersanylargenumbercanbeused, since for largemasses the actual valuemakes little difference;we used 1010 g/mol (wecheckedthatusing104g/molgivespracticallythesameresults,butifthemolarmassisknownitisobviouslybesttousethetruevalue).Notethatcmeltonlyappliestosolutes,sonoapriorivaluesareneeded.The fitting was performed by steepest-descent optimization of δD, δP, and δH, combined withstepwiseadjustmentofr,cmelt,andtheδHD:δHAratio.However,duetotheBooleannatureofthedata(completevs.incompletedissolution)theoptimizedfunctionhasdiscontinuousderivatives,making robust optimization difficult. To overcome this, we performed optimizations from aconcentricgridof129startingpointsaround theaverageofallgoodsolventsandselected thesolutionwith the best fit, usually leading to better fits thanwith theHSP software. In case ofperfect fits, theaveragewastakenofallperfectsolutions found.For faircomparison, thesamealgorithmwasusedbothfortestingtheoriginalHansenmethodandourimprovedmethodology.NotethatthefittingproceduresforHansenspheresarethemselvesasubjectofdiscussion.[25-27]Moreover,theusualdefinitionoftheoptimumisproblematic:whentwodatapointsaremisfittedat opposite sides of the Hansen sphere, any solution between these two points get the samescore. Improvementsofthefittingprocedureitselfshouldalsofocusonthis lineardefinitionoftheoptimum,butthisisoutsidethescopeofthiswork.Moredetailsofourfittingprocedureaswellasthescriptforimplementingitareincludedinthesupportinginformation.



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Thequalityofthefit,qF,wascalculatedasproposedbyHansen:[19]Fordatapointsatthewrongsideofthesphere,theerroriscalculatedbycalculatingthedistanceofthispointtotheedgeofthesphere(oractuallybycalculatingthedifferencebetweentheinteractionparameterβandthevalue for β that would place the data point exactly on the edge of the sphere). Then, qF iscalculatedasfollows:

(20)Here,nisthetotalnumberofdatapointsinthefit(notonlytheill-fittedpoints).Aperfectfitgetsavalueof1;non-perfectfitsgetlowervalues.Notethatthevaluesarecorrectedfortheabsolutescale of the radii, where rsolvent is any solvent’s radius calibrated for the original method andr’solvent is the equivalent radius calibrated for a given setof improvements. In fact, for the fullyimprovedmethodthecalibratedscaleisverysimilartothatoftheoriginalmethod.RESULTSToestablishthepowerofHansen’soriginalmethodandourimprovedmethod,weassessedboththeability to fitknowndataaswell as thequalityofpredictions forunseensolventsbasedonthesefits.Also,therelativeimportanceofthevarioustheoreticalimprovementswasassessedbytesting subsets, where the numbers refer to the numbering in the theory section. Note thatimprovements#3 and#5 are effective onlywhen improvements#1or#2 areused. Similarly,improvement#4onlymakessenseincombinationwith#3.Figures3and4showtheabilitytofitknowndata, expressedby thepercentageofdatapoints that couldnotbe fitted at the correctsideofthesphere,andbythefitquality,qF.Althoughtheimprovementinthenumberofmissedpointsismodest,qFimprovesimpressively,whichmeansthatpointsthatcouldnotbefitted,stillareapproachedmuchcloser.Sincehighfitscoresmayalwaysbecausedbyoverfitting,[28]themostimportantresultsarethepredictionsofthesolubilityinunseensolventsandsolventblends(Figure5).Thesearestronglyimproved aswell. Interestingly, improvements #1 and #5 apparently are themost important.ThecombinationofonlythesetwoimprovementsalreadyleadstoaqFof0.96and77%correctpredictions.WithrespecttotheδDcorrectionfactor,wenotethat,whilethepercentageofmisseddata points during fitting becomes better when the factor is included, the quality of thepredictions actually gets worse. Interestingly, we found that the predictions are even moreimprovedforsolventblendsthanforpuresolvents.However,sincethisreferstodifferentdatapoints,suchnumberscannotbecompared.Therefore,wedidnotanalysethedifferencebetweenpuresolventsandblends.Overall,ourcompleteimprovedmethod(improvements#1+2+3+4+5)leadstoanimprovementoftheqFgoingfrom0.89forHansen’soriginalmethodto0.97,whichisalmostfourtimesclosertoperfectfitting(qF=1).Moreover,thepercentageofcorrectpredictionsimprovedfrom54%to78%. Realising that random predictions will also be correct for 50% of the points, theimprovementislarge.

!F = !! !!!""#"!

!!! !! !solvent!’solvent



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Figure 3. Numbers of missed data points during fitting of 21 training sets with our improvedmethod versus Hansen’s original method, comparing the effect of separate improvements.#1+2+3+4+5denotesthefullyimprovedmethod.δDfactor=2referstothecorrectionfactorHansenusesfortheδDvalues;ourtestswereperformedwithandwithoutthisfactor.

Figure 4.AverageqF (fitquality) for21trainingsetswithour improvedmethodversusHansen’soriginalmethod.PerfectfitswouldgiveqF=1.

Figure5.Resultsforthepredictionofsolubilityatroomtemperatureinunseensolventsandsolventblends (seeMethods section) with our improvedmethods versus Hansen’s originalmethod. Notethat,duetothebooleannatureofthedata,50%correctpredictionsmeanszeropredictivevalue.



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PredictingthesolubilityatextrapolatedtemperaturesThe dependences of the solubility parameters, including r and cmelt, on the temperature, allowmaking predictions at extrapolated temperatures. Thiswas tested by predicting solubilities atlowered temperatures based on experimental tests at room temperature. For comparison, thesolubilityinthestandardtestsetof48solventswasalsomeasuredattheloweredtemperatureforpredictionswithoutextrapolation.Notethatthepercentagesofcorrectpredictionsarelowerthan in Figure 5. This is because only a few of the standard solvents dissolve the solutes atloweredtemperature,resultingininaccuratefittedspheres.Althoughthestatisticsforthesetestsarerather limited,Figure6shows that theextrapolatedpredictionsareat leastasgoodas thepredictionsat constant temperature. In fact, theextrapolatedpredictionsarebetter.The latterreflects the fact that at higher temperaturesmore solvents can dissolve a given solute, whichgivesmoreinformationforfittingandthusamoreprecisefit.Notethatthisimprovementfortheextrapolated predictions is statistically relevant, while the difference between the originalmethodandtheimprovedmethodisnotstatisticalrelevantforthis(small)lowTdataset.

Figure 6.Predictionofsolubilityatlowtemperatures(4to−10°C). ‘lowTfromlowT’meansthattrainingsetsandpredictionsetsareobtainedatthesametemperature. ‘lowTfromroomT’meansthat the same data set is predicted from training sets obtained at room temperature, using theopportunitytoextrapolatethetemperatureintheimprovedmethod.DISCUSSIONMethodsforpredictingsolubilityaretypicallyverygeneralizedinordertomakepredictionsfornewsolutesandunseensolvents.Ontheotherhand,enoughaccuracyisneededtobeofpracticalvalue.Weshowthatbyconsideringthethermodynamicsofsolubility,simplegeneralizedmodelscanbeimprovedsignificantlywithoutsacrificingthegeneralapplicability.Thatsaid,generalizedmethodsareneverexact.Forinstance,mostsolubilitymodelsassumeregularmixing,butrealityisfullofeffectsthatbreakidealitythatcannotbeaccountedforinatheoreticalframework.Forthisreason,westuck to theHansenconceptof fitting,sincethisshouldcaptureat leastpartofthesenon-idealitieswithoutincorporatingthemexplicitly.In recent years, several other models were used for predicting solubility fully theoretically.Examples are: UniQuac,[29] COSMO-RS,[30] and even a combination of COSMO and Hansen.[31]However,COSMOmodelsonlyaimatimprovingthedescriptionoftheenthalpyterms.Therefore,these methods could also be improved further by including entropy and melting corrections,similarlytowhatwehaveshownhere.Wedonotethatfullytheoreticalmethodswillbehuntedbynon-idealitiestoalargerextentthanaprocedureoffittingtorealexperimentaldatasuchasthe Hansen methodology. Hansen parameters can also be estimated theoretically, as studiedextensivelybyPanayiotouetal.[32,33]Theyevenalready includedestimates foracidicandbasiccomponents.[34]Indeed,parameterscalculatedwithPanayiotou’smethodcaninprinciplebeusedinourmethodwhenexperimentalsolventparametersarenotavailable.



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Forourdatasetwithverydifferenttypesofsolutes,weimprovedthequalityofthepredictionsenormously compared to Hansen’s original method. Therefore, in retrospect, one might besurprisedthatHansen’smethodworked in the firstplace for thepaintandpolymer industries.The simple reason is that for polymers the solute solubility radii are very small, utterlyovershadowing the importance of the solvent radii, and thus corrections #1, #2 and #5 haveessentiallynoeffect.Forsmallsolutemolecules,however,thesecorrectionsarehighlyimportant,as shown by our results. Nevertheless, there is still room for improvements. A very practicalpointisthatHansen’sdefinitionofthebestfitisilldefinedduetobeingbasedonlinearerrors,asalreadydiscussedintheMethodssection.Asaresult,thefittingprocedurelackssomerobustnessandourpredictionsmaynotyetbeoptimal.Moreover,ourimplementationofdonorandacceptorparametersisstill imperfect.Manyofthesolutes and solvents in our data set havemultiple functional groups and their parameters arenowaveragedintoonedonorandoneacceptorparameter.Ontopofthat, largemoleculeswithone strong donating group or small molecules with a weaker donating group have the sameparametervalues,whiletheirinteractionswithacceptormoleculesmaydifferdependingonthepropertiesoftheacceptor.Eventhoughfittingofaverageddonorandacceptorparameterswasshowneffectivebefore,[35]morecorrectwouldbetoworkoutallhydrogenbondingpairs fromboth molecules. But implementing that into a solubility parameter approach is notstraightforward.

Another approach to solubility predictions is the work of Ruelle and Huyskens.[36,37] Theyintroduced a very interesting approach to calculate entropy effects that are influenced bypreferentialinteractionssuchashydrogenbonds.ImplementingtheirtreatmentofentropyintoourimprovedHansenmethodologywouldprobablygiveafurtherimprovement.However,todothat, we would again have to let go of the averaged handling of multiple groups, soimplementationintoourapproachisagainnotstraightforward.

A final opportunity to further improve our methodology would be to add a temperature-dependentBoltzmannweighingtermtothepolarandhydrogenbondinginteractions,asalreadydiscussed in theTheorysection.Nevertheless,with the improvementsproposed in thecurrentworkwealreadyimprovestronglyupontheoriginalHansenmethod,aswitnessedbytheresults.It should be noted, though, that these improvements are based on the thermodynamics ofdissolution. Other applications of Hansen parameters, such as swelling, permeability, or gelformationhavedifferentthermodynamics,sodifferentcorrectionsmayapply.

CONCLUSION

The philosophy of Hansen solubility parameters is correct in using parameters for the threefundamental chemical interactions. However, the originalmethodology lacks thermodynamics,making it less suitable for non-polymeric solutes. Here, we have made Hansen’s methodthermodynamically sound by adding simple corrections based on the entropy and enthalpy ofmixingandmelting.

Our improvedmethodologywas tested on an industrial dataset of 15 solutes of very differenttypes,forwhichthesolubilitywaspredictedinunseensolventsandsolventblends.ComparingtoHansen’s original method, the percentage of correct predictions for this dataset improvedimpressively from 54% to 78%. The most important corrections leading to this largeimprovementwere found tobe the introductionof solvent solubility radii anda correction forthe “melting” of solid solutes. Also split donor and acceptor parameterswere applied, but thesuccess of this was somewhat hampered by the necessity to average over several functionalgroupspermolecule.

Additionally,wehave shown thatwith the improvedmethodologypredictions canbemade atextrapolatedtemperatures.Allinall,wehavemadealargestepinimprovingHansen’ssolubilityparameter approach for solubility predictions of polymers aswell as non-polymeric solutes inunseensolventsandsolventblends.Supporting Information: InstructionsforestimatingHbonddonorandacceptorparameters,aderivation of mixing rules for solvent blends, details of our fitting procedure, and full Matlabscriptsimplementingourmethodologyforfreeusage.



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