MultivariateAnalysisofFactorsRegulatingtheFormationofSyntheticAllophaneandImogoliteNanoparticles

McNeillJohnBauer

ThesissubmittedtothefacultyoftheVirginiaPolytechnicInstituteandStateUniversityinpartialfulfillmentoftherequirementsforthedegreeof

MasterofScience

InGeosciences

FrederickM.Michel,ChairFengLin

ClémentLevard

August1,2019Blacksburg,Virginia

Keywords:Imogolite,allophane,synthesis,modeling,growth

MultivariateAnalysisofFactorsRegulatingtheFormationofSyntheticAllophaneandImogoliteNanoparticles

McNeillJohnBauer

Abstract

Imogoliteandallophanearenanosizedaluminosilicateswithhighvalueinindustrialandtechnologicalapplications,howeveritremainsunclearwhatfactorscontroltheirformationandabundanceinnatureandinthelab.Thisworkinvestigatedthecomplexsystemofphysicalandchemicalconditionsthatinfluencetheformationofthesenanominerals.Samplesweresynthesizedandanalyzedbypowderx-raydiffraction,insituandexsitusmallanglex-rayscattering,andhigh-resolutiontransmissionelectronmicroscopy.MultivariateregressionanalysiscombinedwithlinearcombinationfittingofpXRDpatternswasusedtomodeltheinfluenceofdifferentsynthesisconditionsincludingconcentration,hydrolysisratioandrate,andAl:Sielementalratioontheparticlesizeoftheinitialprecipitateandonthephaseabundancesofthefinalproducts.ThedevelopedmodelsdescribedhowincreasingAl:Siratio,particlesize,andhydrolysisratioincreasedtheproportionofimogoliteproduced,whileincreasingtheconcentrationofstartingreagentsdecreasedthefinalproportion.Themodelconfidenceswere>99%,andexplained86to98%ofthedatavariance.Itwasdeterminedfromthemodelsthathydrolysisratiowasthestrongestcontrolonthefinalphasecomposition.Themodelsalsowereabletoconsistentlypredictexperimentallyderivedresultsfromotherstudies.Theseresultsdemonstratedtheabilitytousethisapproachtounderstandcomplexgeochemicalsystemswithcompetinginfluences,aswellasprovidedinsightintotheformationofimogoliteandallophane.

MultivariateAnalysisofFactorsRegulatingtheFormationofSyntheticAllophaneandImogoliteNanoparticles

McNeillJohnBauer

GeneralAudienceAbstract

Allophaneandimogolitearenanosizedaluminosilicatemineralsandstrongly

controlthephysicalandchemicalbehaviorofsoil.Theyholdpromiseforuseintechnologicalapplications.Innature,allophaneandimogoliteareoftenobservedtogetherinvaryingproportions.Similarly,laboratorysynthesisbyvariousmethodsusuallydoesnotresultinpurephases.Theseobservationssuggesttheyformatthesametime,atawiderangeofsolutionchemicalconditions.Itremainsunclearwhatfactorsdeterminehowandwhenthesephasesforminsolution,whichlimitsourunderstandingoftheiroccurrenceinnatureandthelaboratory.Theobjectiveofthisstudyistounderstandandexplainwhatsolutionchemicalandphysicalconditionscontroltheformationofsyntheticimogoliteandallophane.Wedidthisbyutilizingauniqueapproachwherewesystematicallyvariedstartingconditionsofformationoftheseparticles,andthenusedanalyticandstatisticalmethodstodevelopamodelthatdescribestherelationshipbetweeneachofthestartingconditions–concentration,size,pH,atomicratios,andhydrolysisratios,andhowthoseeffectthephaseabundanceoftheparticles.

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AcknowledgementsIwouldliketothankMarcMichelforguidingmethroughthisprocessandhelpingsupportandshapemyscientificmind.IwouldfurtherliketothankmyproxycommitteemembersNancyRossandFengLinwhosteppedintohelpoutatthelastminuteandprovidedvaluablefeedbackandahelpfuloutsiderperspectiveontheproject.IwouldliketothankClementLevardaswellasDonRimstidtwhoprovideddirectionandpurposetotheproject.Iwouldalsoliketothankmyresearchgroup,Aly,Allie,Ali,andRuiforwelcomingmeinandbeinggreatmentorsandfriends.IwanttoadditionallythankAlliewhotookTEMimagesanddidEDSanalysisforme.IwouldliketothankIGEPandNSFforfundingmyassistantshipandtheproject,respectively.IwouldliketoacknowledgeNanoEarthaswellasVirginiaTechGeosciencesforfosteringapositiveenvironmenttowork.FinallyIwouldliketothankfamilyandfriendswhosupportedmethroughoutthiswork.

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TableOfContentsAbstract...............................................................................................................................iiGeneralAudienceAbstract................................................................................................iiiAcknowledgements............................................................................................................ivListofFigures.....................................................................................................................viListofTables......................................................................................................................vii1.Introduction....................................................................................................................12.Methods..........................................................................................................................3

2.1Synthesis...........................................................................................................32.2PowderX-RayDiffractionanalysis....................................................................32.3SmallAngleX-RayScattering............................................................................32.4TransmissionElectronMicroscopy...................................................................42.5MultivariateLinearRegressionAnalysis...........................................................42.6Statisticalinterpretation...................................................................................4

3.Results.............................................................................................................................53.1Synthesesofnanosizedaluminosilicate............................................................53.2Phaseidentification..........................................................................................63.3Linearcombinationfitting.................................................................................73.4Multivariateregressionmodelingofdataset1................................................83.5InsituDv(R)particlesizedata.........................................................................113.6Multivariateregressionmodelingofdataset2..............................................12

4.Discussion.....................................................................................................................154.1Growthstagesofaluminosilicatenanoparticles.............................................164.2Influenceofsynthesisconditionsonphaseabundance.................................164.3Evidenceofallophaneversusproto-imogoliteproductionintheliterature..174.4ModeledversusexperimentalpXRDpatternsproducedinliterature............174.5Thesignificanceoftheinsitudata.................................................................194.6Multivariateapproachtoexplaininggeochemicalsystems............................19

5.Conclusions...................................................................................................................20WorksCited...................................................................................................................21

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ListofFigures

Figure1.ExampleofpHchangeobservedduringthefirsthourofsynthesis…………….……5Figure2.pXRDandHR-TEManalysesofthedifferentendmembersusedforLCFanalysis…………………………………………………………………………………………………………………….…...7Figure3.Measuredversusmodeledresultofimogoliteproportionofsynthesisproducts………………………………………………………………………………………………………………….....…9Figure4.Measuredversuspredictedresultofproto-imogoliteproportionofsynthesisproducts……………………………………………………………………………………………………………………...10Figure5.Measuredversuspredictedresultofamorphoussilicaproportionofsynthesisproducts……………………………………………………………………………………………………………………...11Figure6.ArepresentativedatasetgivenshowingthequalityoftheSAXSsample....12Figure7.Residualsoftheimogoliteproportionmodelsforthedatawithoutandwithparticlesizedatacollected...............................................................................................14Figure8.Residualsoftheproto-imogoliteproportionmodelsforthedatawithoutandwithparticlesizedatacollected.......................................................................................15Figure9.ExperimentalXRDpatternofOhashietal.reportedsyntheticallophane,andDuatal.reportedsyntheticimogolite..............................................................................18

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ListofTablesTable1.SynthesisstartingconditionandnormalizedLCFphaseabundances.................8Table2.Dataoutliningthesecondsetofsyntheses,withstartingconditionsofsynthesisproducts,andtheresultingphaseabundance,withtheadditionofDv(R)particlesizedata..................................................................................................................................13

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1.Introduction

Nanosizedaluminosilicatesstronglycontrolsoilphysicalpropertiesandaffectsoilandwaterqualitybycontrollingthetransportandfateofnutrientsandenvironmentalcontaminants.Theyhavesignificantpotentialinindustrialandtechnologicalapplications(Paineauetal.,2018;Thill,2016;Mailletetal.,2011;Iyodaetal.,2014)andarealsothoughttobetheprecursorstohydroussmectiteandothermorecrystallineclayminerals(Harsh,2002).Ofthenanosizedaluminosilicatesobservedinnaturalsoils,imogolite(Al2(OH)3SiO3OH)andallophane(Al2O3·(SiO2)1.3-2·(2.5-3)H2O)arethemostabundant(Parfitt,2009)andtypicallyoccurtogetherinvaryingproportions(Harsh,2002).Despitetheimportanceofnaturalandsyntheticimogoliteandallophaneitremainsunclearhowphysicalandchemicalconditionsduringformationcontroltheirabundance.Thislimitsourabilitytoexplaintheirdistributionsandimpactsinnaturalsoils,aswellasthepotentialtosynthesizepurephaseswithcontrolledpropertiesforuseintechnologicalapplications.

Theprimarydistinguishingfactorbetweenimogoliteandallophaneistheirmorphologies(Abidinetal.,2007).Imogoliteformsasnanotubes2-3nmindiameterandwithlengthsextendingfor100’sofnmtoµm(Ohashietal.,2002;Gustafsson,2001).Allophaneismostoftendescribedasapseudo-sphericalnanoparticle3-5nmindiameterwithahollowinterior,althoughdirectimagesofwell-formedallophanenanospheresremainscarce(SharpandChang,2017).Further,the3-dimensionalityoftheparticleshasbeenquestionedbysomeresearchers,whosuggestthatTEMimagingcannotdefinitivelydifferentiateaspherefroma2-Dstructure(Levardetal.,2012;HemniandWada,1976).Forthisreason,somehavearguedthatwhatisoftenidentifiedasallophanemayinfactbeonlyrooftile-shapedfragmentsofimogolite.Thesehavebeenreferredtoas“proto-imogolite”basedonevidenceindicatingthatinsolution,theyeventuallytransformtoimogolite(Levardetal.,2010).Botharecomposedofacommonlocalstructurecomprisedofgibbsite-likesheetsofaluminumhydroxide(Al(OH3)).Ononesideofthesheet,isolatedsilica(SiO4)tetrahedraoccupythecenterpositionofthesix-memberedringsformedbytheAloctahedra(Levardetal.,2012).Thecurvatureoftheparticlesisthoughttoarisefromstraincausedbythedifferencesinsizeofthesilicatetrahedraversusthealuminumoctahedra(Thilletal.,2017).

Themorphologicaldifferencebetweenimogoliteandallophaneisdrivenbythedegreeofcurvatureintheprecursorparticles,whereoneandtwodirectionsofcurvatureformtubesandspheresrespectively.Whatdeterminesthiscriticalparameterremainsunclear.Previousexperimentalstudiesofimogoliteandallophaneformationhavetestedseparatelytheeffectsofstartingconcentration,pH,andelementalratiosonthetwodifferentmorphologies(Wangetal.,2018;Ohashietal.,2002;Montarges-Pelletieretal.,2005;Duetal.,2018).Astudybasedonnumericalmodelingsuggestedthatthereisacriticalsizetotheproto-imogoliteparticlespreviouslydescribed(Thilletal.,2017).Itpredictsthatparticleslargerthan4nmfavorimogoliteformationwhereassmallerparticlesresultinallophane.Othershavesuggestedthatexcesssilicatoaluminuminthestartingsolutionwillsuppressimogoliteformationduetothelimitedcapacityforsilicapolymersintheinnertube.Excesssilicamaybeaccommodatedby

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silicapolymersresidinginthelargerinteriorsofsphericalallophaneand/orbytheincreasednumberofreactivesurfacesitesonproto-imogolitefragments(Levardetal.,2012).Anotherstudysuggestedthatimogoliteisnotfavoredathydrolysisratioslessthan1.5duetodifferencesinimogoliteandallophanestabilitiesatdifferentpHandOH:Alranges(Levardetal.,2011a).Takentogetherthesepriorstudiessuggestthatimogoliteandallophaneformationmaybeinfluencedbytwoormorecompetingfactors.

Thispaperexplainshowkeyvariablesincludingconcentration,elementalAl:Siratio,hydrolysisratioandspeed,andprecursorparticlesizeimpacttheformationofimogolite.Systematicsynthesisexperimentswerecombinedwithinsituandexsituanalyticalcharacterizationandmultivariatelinearregressionanalysistodeterminethequantitativeinfluenceeachfactorhadonfinalphaseabundance.Wefoundnoevidenceforwell-formedallophanenanospheresatanyofoursynthesisconditions.Instead,onlyimogolitenanotubesandwhatweinterpretasproto-imogolite,whichhasphysicalcharacteristicssimilartowhathasbeendescribedpreviouslybyLevardetal.(2010).Thisisthefirststudydesignedtounravelthecompetingeffectsofthesedifferentsynthesisvariables,aswellastofactorintheroleofprecursorparticlesizeonimogoliteformation.Theresultsareimportantforgreaterunderstandingofaluminosilicatesynthesis,whereindividualvariablescanbeisolatedandunderstoodfortheireffectonthesystem.Theframeworkofthestudycanbeadaptedandutilizedforothergeochemicalprocesseswherecompetingeffectscomplicatethestudyofthesystem.

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2.Methods2.1Synthesis:Nanosizedaluminosilicatesweresynthesizedatroomtemperatureusingamethodadaptedfromtheliterature(Iyodaetal.,2014;Montarges-Pelletieretal.,2005;Wadaetal.,1979;Huangetal.,2016).Aluminumchloridehexahydrate(AlCl3·6H2O,Sigma-Aldrich,99%)wasmixedwith50mLof18.2MΩ·cmultrapurewaterinastandardborosilicateglassreactortoreachconcentrationsrangingfrom0.005Mto0.2M.Tetraethylorthosilicate(TEOS,Sigma-Aldrich,99.9%)wasimmediatelyaddedtoreachAl:Simolarratiosrangingfrom0.5to2.TEOSwasaddedpriortoinducinghydrolysistominimizesilicapolymerization.Hydrolysiswasachievedbypumpinga0.1Msodiumhydroxidesolution(NaOH,FisherScientific,99%)atratesrangingfrom0.2to10mLmin-1,usingaperistalticpump(IsmatecIPC8)fittingwithTygon2-stoptubing(IsmatecSC0824)toachieve[OH]:[Al]hydrolysisratiosof1to3.Theresultingsolutionwasstirredat400RPMfor1hour,whilemonitoringpHat5-secondintervals(OaktonPC2700).Thesolutionswerethentransferredinto250mLanaerobicglassbottleswithbutylrubberstoppersandaluminumseals(ChemglassLifeSciencesCLS-4217-03),andthenheatedconstantlyat95°Cfor7days(FisherIsotemp215G).Thefinalsolutionwasthendialyzedagainst18.2MΩ·cmultrapurewaterusinga12-14,000Da,25mmdiameterstandardgrademembrane(SpectraPor)for2weeks,untiltheconductivityofthesolutionreachedbelow2μS.Thedialyzedsolutionwasthendriedat40°Ctoobtainawhitepowderforfurtheranalysis.Synthesisofanamorphoussilicasampleusedasareferencewasreportedpreviously[Cismasuetal.,2014]2.2PowderX-raydiffractionanalysis:Powderx-raydiffraction(pXRD)datawerecollectedusingaRigakuMiniFlexIIDesktopx-raydiffractometerequippedwithaCuKαsource(30kVto15mA).Driedsampleswerehomogenizedusinganagatemortarandpestleandthepowderswerepackmountedintoawellinazerobackgroundsinglecrystalsiliconholder.Diffractedintensitiesfromthesampleswerecollectedfrom5to80°2θin0.01°increments.Exposuretimewas2secondsperstepandthesampleholderwasrotatedcontinuouslyduringdatacollection.Linearcombinationfitting(LCF,WinXAS3.0)wasusedtoestimatetheproportionsofimogolite,proto-imogolite,andamorphoussilicaineachsynthesizedproduct.pXRDpatternsofimogolite,proto-imogolite,andamorphoussilicawereselectedasendmembercomponentsbasedonmaximumpurityassessedbypXRDcomparisons,SAXS,andTEMdata(seeSection3.2).TernarycombinationsoftheendmemberpXRDpatternswerefittedtoeachsamplepatternandweightfractionsforeachcomponentweredeterminedusingaleast-squaresminimizationprocedure.Thesumoftheweightfractionswasnormalizedto1.0.Residualswerevisuallyassessed.2.3SmallAngleX-rayScattering:

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Smallanglex-rayscattering(SAXS)datawerecollectedusingaPanalyticalEmpyreanNanoEditionmulti-purposex-rayscatteringplatformequippedwithaCuanode(λ 𝐶𝑢!! =1.5406Å)andellipticbeamfocusingopticwitha1/32°fixedslit.Diffractedintensitiesfromthesamplesandbackgroundswerecollectedfrom-0.15to6°2θinapproximately0.014°incrementsusingaGaliPIX3Dareadetectororproportionaldetector.Exposuretimesrangedfrom60-320sec/point.DrysampleswereflatmountedbetweenMylarfilmsandmeasuredinaScatterX78evacuatedsampleholderandbeampath.InsituexperimentswereperformedusingacustomliquidflowcelldesignedforusewiththeScatterX78vacuumstage.Inbrief,thesynthesissolutionwaspumpedcontinuouslyfromthereactorintotheflowcellandthenreturnedtothereactorusingaperistalticpump.Scanswerecollectedcontinuouslyduringeachexperimenttomonitorforchangesinparticlesizedistributionduringinitialhydrolysisandsubsequentmixing.Oncethesolutionbegantoshowstableresults,usuallyupto90minutesintotheexperiment,thefinal5scanswereaveragedandthebackgroundsubtracted.Volume-weightedsizedistribution(Dv(R))analysis(seeGlatter,1980)wereperformedusingEasySAXS(Panalytical).2.4TransmissionElectronMicroscopy:High-resolutiontransmissionelectronmicroscopy(HR-TEM)imagingwasperformedonaJEOL2100operatingat200KeV(NCFL,ICTAS,VirginiaTech).Samplesweredilutedto10ppm,andthendialyzedagainstmethanolusinga12-14,000Da,25mmdiameterstandardgrademembrane(SpectraPor).Beforeimaging,sampleswereplacedinasonicatingbathfor1minutetodisperseaggregates.ThesamplesweremountedonLacey-carbon,300mesh,copper-backedgridsfromTedPella.Thegridsweredriedinairandstoredinabenchtopvacuumchamber.2.5MultivariateLinearRegressionAnalysis:MultivariatelinearregressionanalysiswasappliedtoLCFdatathatestimatedeachendmemberproportionofimogolite,proto-imogolite,andamorphoussilica.Thestartingphysicalandchemicalconditionswereanalyzedusingmultivariateregressionwithrespecttothefinalphasecompositionofthesynthesizedproducts.InthecasewhereadependentvariableYcanbedescribedbyarelationshiptosomenumberofindependentvariablesxibysomefunctionY=f(xi),multivariateregressionanalysescanbeemployedtodeterminetheexactnatureoftherelationshipofeachindependentvariabletothedependentvariable,whilesimultaneouslycontrollingforalloftheotherindependentvariables.(Mansourietal.,2018).ThisanalysiswasperformedusingtheRsoftware,RStudio.2.6Statisticalinterpretation

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Whenanalyzingtheeffectivenessofthemultivariatemodels,R2representsofhowmuchofthevarianceinthedataisexplainedinthemodel.BecauseeachindependentvariableconsequentlyincreasestheR2value,anadjustedR2calculationmustbeperformed.ThisaccountsfortheincreaseinR2withadditionalvariables,andgivesatruerepresentationofhowmuchofthedataisexplainedbythemodel.

Therootmeansquareerror(RMSE)isthestandarddeviationoftheresidualofthedata.Itgivesanestimationofpredictivepowerofthemodel. DurbinandWatson(DW)statisticswereusedtotestforautocorrelationintheresidualsfromtheregressionanalysis(DurbinandWatson1950,1951).Randomresidualsareacrucialcomponentofapredictivemodel,correlationintheresidualsindicatessomepredictiveinformationisnotdescribedbythemodel.TheDWstatistic(d)alwaysliesbetween0and4.Thenullhypothesisisconfirmedwhendliesbetween1.5and2.5,whilehigherandlowervaluessuggestautocorrelation.3.Results3.1Synthesesofnanosizedaluminosilicates

Figure1ExampleofpHchangeobservedduringthefirsthourofsynthesis,brokenintothreegrowthstages.Thissynthesiswasperformedusing0.05MAlCl3witha1:2Al:Si

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ratio,anda1:1OH:Alratio.TimepointzerocorrespondswiththestartingpHandthebeginningofNaOHaddition.

ApHandvisualstudyofthesynthesisprocesssuggestedthreedistinctgrowthphasesasdisplayedinFigure1.NaOHadditionduringtheinitialminutesoftheexperiment(Stage1)resultsinarapidriseinpH.ThepHbeginstodecreaseassoonasNaOHadditionisfinished,whichisapproximatelyatthemaximumbetween0-5min.Duringthisperiodatranslucentgel-likephaseappearsinthesolution.Stage2ismarkedbyaperiodofrapidpHdecreasewherethesolution,dependingonstartingconditions,eitherreturnstoclearordevelopsawhitecloudyprecipitate,whichpersistsfortherestofthesynthesis.Stage3istheperiodinwhichthepHbecomesapproximatelystable.Thesolutionwasthenplacedinanaerobicbottlesandheatedfor7daysat95C,afterwhichacloudywhiteprecipitatewouldoftenform.Whendried,thesolutionwouldproduceafinewhitepowder.3.2Phaseidentification

ThefinalsolidsofeachsynthesiswerecharacterizedbyHR-TEMimagingand/or

pXRDanalysis.Threedistinctphaseswereidentified.Figure2presentspXRDprofilesforsamplesconsistingofpureimogoliteandproto-imogolite,aswellasforanamorphoussilicareference.HR-TEMimagesofimogoliteandproto-imogolitearealsogiveninFigure2,alongwithanexampleofamorphoussilicaobservedinoneofthesynthesisproducts.AllthreepXRDprofilesshowaseriesofoneormorebroad,weakpeaksconsistentwithmaterialslackinglong-rangeperiodicity(i.e.,short-rangeordered).ThepXRDpatternsofeachagreewithpreviouslyreportedspectra(Arancibia-Mirandaetal.,2013;Levardetal.,2012;Musićetal.,2011).HR-TEMimagesoftheimogoliteandproto-imogoliteendmembersgenerallyshowedhighlyaggregatednanoparticleswithdifferentmorphologies(Figure2AandB).Theimogolitesampleexhibitedfairlydistinctandelongatednanotube-likeshapes.ThesamplethatwasexpectedtobeallophanebasedonitspXRDcharacteristicsshowednoevidenceofwell-formedsphericalnanoparticles.Thus,weassumedthatthemorphologicalcharacteristicsofthissamplewereconsistentwithproto-imogolite.Amorphoussilicaconsistedofaggregatesofglobularwithvaryingparticlesizesintherangeof10’sto100’sofnms(Figure2C).ThiswasconfirmedthroughEDSanalysiswhichshowedthesamplecontainedonlysilicon(needtoaddthistomethodsnowbutnotsureonspecsyet).Overall,thesizesandmorphologiesobservedinHR-TEMweregenerallyconsistentwithwhathasbeenreportedpreviouslyforsyntheticimogolite,proto-imogolite,andamorphoussilica.

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Figure2pXRDandHR-TEManalysesofthedifferentendmembersusedforLCFanalysis.RepresentativeTEMimagesofA)amorphoussilicaB)imogoliteandC)proto-imogoliteendmembers.Italsoincludesasynthesizedproductshownindarkgrey,andtheresultingLCFfitinlightgray. Themajorityoftheotherproductsofthedifferentsynthesisexperimentsdidnotresultinapuresinglephaseascommonlyfoundinnaturalsamples,(Harsh,2002),butinsteadcontainedamixtureofthethreedifferentendmembers.QualitativeevidenceforthesemixtureswasprovidedbypXRD,whichshowedcombinationsofdistinctpXRDpeaksforimogolite,proto-imogoliteand/oramorphoussilicaallpresentinasinglesample(Table1).

3.3LinearcombinationfittingLinearcombinationfittingofpXRDwasusedtoquantifytheabundancesof

imogolite,proto-imogolite,andamorphoussilicainasetof23synthesisproducts.AnexampleoftheLCFfitofapXRDpatternfromasynthesizedproductisshowninfigure2.Table1showstheresultsofLCFanalysisandsummarizesthestartingconditionsusedforeachsynthesis.LCFresultsshowthattheabundancesofproto-imogoliteandimogolitevariedfromapproximately0upto100%forthedifferentsynthesisconditions.Theabundanceofamorphoussilica,whichhasbeenreportedpreviouslyinstudiesofimogoliteandproto-imogolitesynthesis(Wadaetal.,1979),variedbetween

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approximately0upto31%.Thedataillustratehowvariationsintheabundancesofproto-imogolite,imogolite,andamorphoussilicaoccurwithdifferencesinsynthesisconditions.Table1SynthesisstartingconditionandnormalizedLCFphaseabundances

3.4Multivariateregressionmodelingofdataset1

ThedatainTable1wasanalyzedusingmultivariateregression,whichproducedamodelofimogoliteproportionthatfollowedthefollowingrelationships:I = 0.009+ 0.35hydro−1.82c+ 0.11Al : Si [Eq.1].Themodelwasusedtoproducedataofpredictedimogoliteproportion,whichwascomparedtotheexperimentallydeterminedproportionsinFigure3.

Concentration(M)

HydrolysisRatio

NaOHaddition(ml/min)

Al:SiRatio

Proto-imogoliteProportion

ImogoliteProportion

AmorphousSilicaProportion

0.15 3.00 10.00 1.50 0.00 1.00 0,000.13 3.00 10.00 1.50 0.00 1.00 0.000.13 1.00 2.00 1.50 0.66 0.12 0.220.10 3.00 10.00 1.50 0.05 0.95 0.000.18 1.00 2.00 1.50 0.64 0.12 0.240.13 1.00 2.00 1.00 0.75 0.29 0.000.15 1.00 2.00 1.00 0.73 0.10 0.170.01 3.00 2.00 1.50 0.00 1.00 0.000.02 1.00 10.00 1.50 0.48 0.33 0.190.15 3.00 10.00 1.50 0.00 0.78 0.230.18 3.00 10.00 1.50 0.32 0.49 0.190.18 1.00 2.00 1.50 0.62 0.08 0.290.18 3.00 10.00 1.50 0.62 0.24 0.140.18 1.00 2.00 1.00 0.29 0.15 0.560.20 1.00 2.00 2.00 0.74 0.00 0.270.18 1.00 2.00 2.00 0.79 0.00 0.240.13 1.00 10.00 2.00 0.53 0.14 0.340.19 1.00 2.00 1.00 0.62 0.03 0.350.19 1.00 2.00 2.00 0.48 0.06 0.460.16 1.00 10.00 1.50 0.58 0.13 0.290.16 1.00 10.00 1.00 0.58 0.01 0.400.16 1.00 2.00 2.00 0.60 0.20 0.200.20 1.00 10.00 1.50 1.00 0.00 0.00

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Figure3Measuredversusmodeledresultofimogoliteproportionofsynthesisproducts ThemodeldemonstratesthatimogoliteproportionincreaseswithincreasinghydrolysisratioandAl:Siratio,andthattheproportionofimogolitedecreasesasconcentrationofstartingreagentsincreases.Themodelwasfoundtobesignificant,withap-valueof1.3x10-9,andanR2of0.86,whichmeansthatthemodelexplains86%ofthevarianceinthedata.Thestartingconcentrationandhydrolysisratiofactorswerebothstatisticallysignificant,with>99.99%and>99%confidencerespectively.Boththeinterceptandelementalratiofactorbothfaileda0.05pvaluesignificancetest.

TheRMSEis0.13,meaningthemodelcanpredicttheimogoliteproportiontowithin±13%.TheDWstatisticfortheresidualsofthismodelis2.26,whichwitha99%confidenceratethenullhypothesiscanberejected,suggestingthereisnoautocorrelationintheresiduals.

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Figure4Measuredversuspredictedresultofproto-imogoliteproportionofsynthesisproducts

Amodelfordeterminingtheproto-imogoliteproportionwasalsodeveloped,andisdescribedasfollows:PI =1.01− 0.30hydro+ 0.93C − 0.092Al : Si [Eq.2] Themodeldescribesthatasconcentrationofstartingregentsincreases,theproportionofproto-imogoliteinfinaltheproductalsoincreases.Italsosuggeststhatasthehydrolysisratioandelementalratioincreasetheproto-imogoliteproportionisdecreased,directlyopposedtotheimogoliterelationships. Thismodelwasfoundtobesignificant,withap-valueof1.62x10-9.IthasanR2of0.88;themodeldoesnotexplain12%ofthevarianceofthedata.Theinterceptandhydrolysisratiofactorwerefoundtobesignificantwitha>99.99%confidence.Theconcentrationfactorwasalsofoundtobesignificanttoapvalueof0.05.Aswiththeothermodel,theelementalratiowasinsignificant,withap-valueof0.22.TheRMSEissimilartotheimogolitemodel,butslightlylowerat0.12.TheDWstatisticwasfoundtobe0.5,whichrejectsthenullhypothesisandsuggestspositiveautocorrelationintheresiduals,meaningthereissomepredictivemeasureinthedatathatisnotcapturedbythemodel.

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Figure5Measuredversuspredictedresultofamorphoussilicaproportionofsynthesisproducts

Amorphoussilicawasfoundtohavesignificantvarianceandlimitedpredictivepower,andsoafullmodelisnotreported.Themodelis,basedonp-values,9ordersofmagnitudelesssignificantthantheproto-imogoliteandimogolitemodels.Theoverallsignificanceis99%,buttheaverageconfidenceofalloftheindividualfactorsis65%.Noneofthevariableswerefoundtobesignificant.Intuitivelyhowever,theAl:SiratiosuggestedthatasSiincreasesrelativetoAl,totalamorphoussilicaproportionincreasesat85%confidence.Thep-valueofthemodelis0.01,buttheR2valueis0.32,meaningthemodelexplainsonly32%ofthevarianceoftheamorphoussilicaproportion.TheDWstatisticfortheresidualsis0.68,indicatingpositiveautocorrelation.3.5InsituDv(R)particlesizedata

WeperformedinsituSAXSstudieswherewerepeatedtheoutlinedsynthesisproceduresandmeasuredaverageandmedianparticlesize(Figure6).Wehaveconfirmedthatincreasingconcentrationsofstartingreagentsreducestheaverageprecursorparticlesize,followingequation:Ravg = 2.3− 9.6C − 0.88s+ 0.68H − 0.33Al : Si [Eq.3]

−0.1

0.0

0.1

0.2

0.3

0.0 0.1 0.2Predicted AmSi Proportion

Mea

sure

d Am

orph

ous

Silic

a Pr

opor

tion

In situ Amorphous Silica linear model

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ThismodelexplainsaverageprecursorparticlesizewhereCisstartingconcentrationofreagents,sisspeedofNaOHaddition,Hishydrolysisratio,andAl:Siiselementalratio.Themodelexplained91%ofthevarianceinthedata,withaconfidence>99.99%,andhadrandomresidualsasevidencedbyaDWstatisticof2.08.Increasingconcentrationdecreasesaverageprecursorparticlesize,andhasthelargestinfluenceonthefinalsizedistributionoftheparticles.

Figure6.ArepresentativedatasetgivenshowingthequalityoftheSAXSsample(Greytoblack),background(Purple)(Top),aswellastheDvRfit,overatimeseriesofdatacollected(Bottom)

TrackingDv(R)resultsinsituthroughthefirsthourofsynthesisyieldedsimilarresultstothepHstudyshowninFigure1,andisshowninFigure6.Threestagesemergedagain,wherestage1showsrapidgrowthandnucleationofparticles,withincreasingaverageparticlesize.Stage2showsminimalgrowthinparticlesizebutcontinuednucleation,andbystage3theDv(R)analysisshowsnogrowthovertimeinsizeorsignificantnucleationofparticles3.6Multivariateregressionmodelingofdataset2

Asecondsetofsyntheseswasperformedinsitu,andusingDv(R)analysisaverageandmostfrequentparticlesizedatawereaddedtothesystem.Theseexperimentsaimedtoquantifythesizesoftheparticlesthatwouldbeheatedand

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agedtoobtainthefinalsynthesisproducts.Oncestage3wasreached,andtherewasnolongerevidenceofnucleationorgrowth,theparticlesizedatawasrecorded,andaddedtothesetofsynthesisconditions(table2).Thedatawasthenanalyzedidenticallytothepreviousdatasettoproduceproportionabundancemodels.Theimogoliteproportionwasmodeledasfollows:I = −0.046+ 0.34hydro− 0.40C − 0.073Al : Si+ 0.11Ravg − 0.38Rmed [eq.4]Table2Dataoutliningthesecondsetofsyntheses,withstartingconditionsofsynthesisproducts,andtheresultingphaseabundance,withtheadditionofDv(R)particlesizedata

Themodeldemonstratesthatasaverageparticlesizeincreases,theimogoliteproportionalsoincreases,andthattheinverseistrueforthemedianparticlesize.Thehydrolysisratioconstantwascalculatedto>99.99%confidence,theelementalratioconstantwascalculatedto99%confidence,andtheconcentrationandmeanradiussize

Concentration(M)

HydrolysisRatio

NaOHaddition(ml/min)

Al:SiRatio

MeanRadius(nm)

MedianRadius(nm)

Proto-imogoliteProportion

ImogoliteProportion

AmorphousSilicaProportion

0.1 1 5 1 0.73 0.70 0.86 0.15 0.000.1 1 2 1 0.80 0.75 0.82 0.12 0.050.15 1 1 2 0.93 0.70 0.75 0.25 0.000.15 1 5 2 0.57 0.56 0.79 0.21 0.000.15 1 0.5 2 0.87 0.90 0.85 0.08 0.070.15 1 2 2 0.78 0.80 0.71 0.19 0.100.1 1 0.5 2 0.90 0.90 0.58 0.26 0.160.175 1 5 2 0.75 0.80 0.81 0.11 0.070.175 1 2 2 0.82 0.80 0.63 0.23 0.140.175 1 5 1 0.66 0.60 0.63 0.11 0.260.15 1 5 1 0.79 0.75 0.57 0.12 0.310.175 1 0.5 2 0.82 0.80 0.56 0.15 0.280.125 1 1 2 0.91 0.90 0.85 0.03 0.120.175 1 0.5 1 0.84 0.75 0.64 0.10 0.270.125 1 5 2 0.73 0.70 0.57 0.31 0.120.125 1 2 2 0.85 0.80 0.63 0.22 0.150.1 3 5 2 1.98 0.80 0.00 1.01 0.000.01 3 2 2 3.63 1.40 0.00 1.00 0.000.005 3 2 2 4.10 1.40 0.00 1.00 0.000.005 3 2 1 4.32 1.60 0.00 1.00 0.000.2 3 2 2 2.76 1.00 0.11 0.89 0.000.15 3 2 2 1.82 0.80 0.00 1.00 0.000.05 3 5 2 1.90 0.80 -0.07 1.00 0.00

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constantswerecalculatedto>98%confidence.Thep-valuefortheentiremodelis9x10-14.Theinterceptofthemodelandthemedianparticlesizewerefoundtobestatisticallyinsignificantwithap-valueof0.15and0.07respectively.TherateofNaOHadditionwasfoundtodecreasetheoverallconfidenceofthemodel,withap-valuethatsuggestedadditionratedidnotinfluenceendmemberproportion,andsowasomitted.TheadjustedR2ofthemodelis0.98.TheRMSEofthemodelis0.05,meaningthatthemodelcanpredicttheimogoliteproportiontowithin±5%.

Thethreeoverlappingfactorsforbothequations1and4(concentrationAl:Siratio,OH:Alratio)sharedidenticaldirection,andthecoefficientswereallwithinastandarderrorofoneanother.ThesimilarityofthemodelsisdescribedwellbytheresidualsofthedatashowninFigure7.TheDWstatisticforequation4was1.84,whichfailedtorejectthenullhypothesisat99%confidence.Visually,theredoesappeartobealineartrendintheresidualsatlowerproportionsofimogolite,withbothmodels.

Figure7Residualsoftheimogoliteproportionmodelsforthedatawithout(red)andwith(black)particlesizedatacollected.

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Figure8Residualsoftheproto-imogoliteproportionmodelsforthedatawithout(red)andwith(black)particlesizedatacollected.

Theproto-imogoliteproportionofthisseconddatasetwasmodeledby:PI =1.08− 0.34hydro+ 0.37C − 0.0007Al : Si− 0.034Ravg + 0.050Rmed [Eq.5]. ThismodelsharesthesametrendsasthemodeldevelopedthroughthedatainTable1,andtheabsolutefactorsarewithinastandarddeviation.Themodeldescribesrelationshipsthatareoppositethoseoftheimogoliteproportion,includingtheparticlesizewheremeanparticlesizegrowthleadstolessproto-imogolite.

Thep-valueoftheoverallmodelis3x10-8,alongwithanR2valueof0.91,indicatingahighdegreeofconfidencethatthemodelisexplaining91%ofthevarianceinthedata.Unliketheimogoliteproportionmodel,theindividualvariablesarelesssignificantthanwhencombinedintothegeneralmodel.Thecoefficientfortheintercept,hydrolysis,andconcentrationweresignificantto>95%,buttherestofthefactorsrangedfromp-valuesof0.1to0.8.TheRMSEofthismodelisalsohigherthanthatoftheimogolitemodel,at0.09;thismodelcanpredicttheproto-imogoliteproportion,onaverage,towithin9%.

TheDWstatisticfortheresidualsofthismodelis0.5,whichconfirmsthealternativehypothesisthatthereispositiveautocorrelationintheresiduals.Visualinterpretationoftheresidualsshowninfigure8alsoshowsboththesimilarityofthemodelsandthelinear,predictabletrendathighconcentrationsofproto-imogolite.4.Discussion

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4.1Growthstagesofaluminosilicatenanoparticles

ResultsfrompriorstudiesconductedatsimilarconditionsuggestthatStage1

involvestherapidnucleationandgrowthofgibbsite-likeAl(OH)3nanoparticles(Wadaetal.,1979;Ohashietal.,2002)(Figure1).Stage2isthoughttoinvolvesilicaattachmenttothesheetsresultingintheformationofthelocalimogolitestructure(Duetal.,2018).Thisoccursthroughtheoxolationmechanismwheresomeofthe–OHgroupsonthegibbsite-likesheetsaresubstitutedbythesilicatetrahedra,releasing3molesofH3O+intosolutionpermoleofofSiattached.(Arancibia-Mirandaetal.,2013).ThestabilizationinpHandparticlenucleationandgrowthwithtimesuggeststhatthedevelopmentoftheimogolitestructureslowsandeventuallyceasesduringStage3.4.2Influenceofsynthesisconditionsonphaseabundance

Themodelsdevelopedinequations1-5describetherelationshipbetweenthesystemofphysicalandchemicalconditionsusedforsynthesis,andtheabundancesofimogolite,proto-imogolite,andamorphoussilicathefinalsynthesizedproducts.

Thehydrolysisratiowasdeterminedtobethemostimportantfactorcontrollingmorphology.Asthehydrolysisratioincreasedfrom1to3,theproportionofimogoliteincreased,andtheproportionofproto-imogoliteformeddecreased.AconsequenceofincreasingthehydrolysisratioisthattheinitialpHofsolutionalsoincreases.Withahydrolysisratioof1,thepHrangedfrom3.3-4.2,whilewhenthehydrolysisratiowassetto3,thepHrangedfrom9.2to10.Allophaneshavebeenshowntobeunstableinalkalineconditions,wheredefectporesdevelop,enlarge,andbreakdownthestructure(Wangetal.,2018).Asimilarmechanismwouldlikelypreventproto-imogolitesfromforminginalkalineconditions,explainingthedecreaseofproto-imogoliteproportionwithincreasinghydrolysisratio.

Previousstudiesofimogolitesynthesishavefoundthatathydrolysisratios

below1.5,tubularstructureswereunabletoform,andsignificantstructuraldefectsbelow2.0wereobserved(Levardetal.,2011b).Theimogolitemodelsupportsthatexperimentalresult;adecreasinghydrolysisratioresultedinsignificantlydecreasedimogoliteproportion.Experimentally,theaverageimogoliteportionofsampleswithahydrolysisratiobelow1.5was9%,whichagreeswellwiththepreviouslyreportedfindings.

TheAl:Siratiowasalsofoundtobeasignificantfactorthatinfluencesfinalphasecomposition.AstherelativeabundanceofAlincreases,theimogoliteproportionincreases.TheidealizedformulaforimogoliteisAl2(OH)3SiO3OH,witha2:1Al:Siratio.Astheratioisdeviatedfrom,andtheSiincreases,theamountofamorphoussilicaincreasesaswell.Thisamorphoussilicalikelyinterfereswiththegrowthkineticsofimogolite.Proto-imogoliteparticleshavebeenshowntobeabletoincorporatesilica

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intoitsstructure,polymerizingchainsofSibranchingofffromthetetrahedralsites,creatingaSi-richlocalstructure(Levardetal.,2012).Thisisreflectedinthemodelequations,whereastherelativeamountofSiincreases,theproto-imogoliteproportionalsoincreases.

Themodelsshowthatincreasingconcentrationofstartingreagentsleadstoalowerproportionofimogolite.Basedonreportedliteratureresults,producinghighconcentrationsofnucleatedparticleshavebeensuggestedtoimpedeimogolitegrowthkinetics,especiallygrowinglongertubestructures(Mailletetal.,2011). Basedonresultsfromnumericalmodeling,ithasbeenproposedthatthereisasizethresholdat4nmforprecursorparticleswheretheenergeticallyfavorablemorphologyforgrowthtransitionsfromsphericaltotubular(Thilletal.,2017).Thisresultagreeswiththeinsitustudiesthataredescribedbeequations4and5.Theprecursorparticlesthatwere>3.6nmindiameterhad,onaverage,anincreaseofimogolitefinalphasecompositionof0.92.Thisconfirmstheresultofthenumericalmodelingexperimentally,withaslightadjustmenttothemorphologicaltransitionto3.6nm.4.3Evidenceofallophaneversusproto-imogoliteproductionintheliterature Extensivesynthesesofnanosizedaluminosilicateshavebeencarriedoutbyavarietyofresearchersatdifferingstartingconditions.Manyofthesestartingconditionsoverlapdirectlywiththosechosenforthisstudy.Althoughtheauthorsofthisstudyneverproducedanywell-formednanospheres3-5nmsizesthatwereexpectedbasedonwhathasbeendescribedpreviouslyforallophane,multipleotherstudiesutilizingsimilarmethodsandconditionsreportallophaneproduction.TheXRDpatternsofallophaneandproto-imogoliteareindistinguishable,andsocannotbeusedasevidencetowardstheproductionofallophane(Levardetal.,2012).HRTEMimagesarethestrongestevidenceforallophaneasadirectmeasureofmorphology,butitisnotaubiquitoustechniqueused,whendistinguishingbetweenimogoliteandallophane.Recentmodelsofallophanicproductscallintoquestiontheidealized3-5nmspheralstructures;becauseithasavariableAl:Siratioandthestructureisnotaswelldefined(Yuanetal.,2016).Ithasbeenproposedthatallophane,becauseofitsvariablecomposition,shouldbethoughtofasaseriesofmineralsratherthanasinglespecies(ParfittandKimble,1989).Itistheopinionoftheauthorsthatoftenwhatiscalledallophaneintheliterature,unlessisclearlydemonstratedthroughTEMimagery,isactuallyproto-imogoliteinavarietyofmorphologies.4.4ModeledversusexperimentalpXRDpatternsproducedinliterature

Thegeneralizabilityofthemodelwastestedbycomparingthemodel’spredictionofphaseabundanceandthereportedpXRDpattern,basedongivenstartingconditions.Whetherornottheresearchersdescribetheproductsas

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allophaneorproto-imogolite,theresultshouldbethesamebecausetheyshareindistinguishableXRDpatterns.Thekeydifferentialisbetweenthoseandimogolite,whichhaslongerrangeorderandadistinctpXRDpattern.Synthesisconditionsreportedinliteraturewererunbythemodels,resultinginpredictedphaseabundancesofimogolite,proto-imogolite,andamorphoussilica.Usingthisresult,anexpectedpXRDpattern,basedontheconditions,wasproduced.ThiswasthencomparedtothereportedpXRDpatterninthestudywherethoseconditionswereused(Figure9).. Ohashietal.(2002)wereattemptingtosynthesizeallophanefromhighconcentrationsolutions,usingstartingconcentrationsof0.1M,aSi:Alratioof0.75,anddidnotusedNaOHtoinducehydrolysis.Despiteusingconditionsbeyondthescopeusedtodevelopthemodel,aswellasaslightlydifferentsilicareagent(Na4SiO4)thereisstillstrongagreementbetweenthepatterns.

Figure9ExperimentalXRDpatternoftheOhashietal.reportedsyntheticallophane,andtheDuatal.reportedsyntheticimogolite.Theexperimentallyderivedpatternsareinblack,andthemodeledpatternsbasedonthegivenstartingconditionsareingrey. Duetal.(2017)investigatedtheeffectsofheatingtimeonimogoliteformation.Theresultsofthefinalproductheatedfor5dayscloselyalignswiththemodeledresultshowninFigure9.Themodelcorrectlypredictstheresultingproducttobeprimarilyimogolite,andalignswiththeexperimentalpattern.Thereisamissingpeakinthemodelat0.94nmthatislikelyduetoahigherdegreeoforderingoftheimogolitetubesinthesamplethatDuetal.produced.ThestartingconditionsthatDuetal.usedwere

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startingreagentconcentrationsof0.15MAlCl3,0.1MNa4SiO4,andaddedNaOHtoreachapHof5.5.Onceagain,usingconditionsnotexplicitlytestedindevelopingthemodel,thereisstillgoodagreementbetweenwhatthemodelsexpecttoresultfromtheconditions,andthereportedexperimentalpattern. Theseexamplesdemonstratehowtheabilityofourmodeltopredicttheoutcomeofthereportedsynthesesfromliteraturereview,evenwhensynthesisconditionsoutsidetheboundsusedtoproducethemodelwereused.Thisconfirmsthatthemodelismorewidelyapplicablethanthespecificsynthesismethodandproductsproducedinthisstudy,andcanprovidevaluableinformationtoresearchersworkingwithnanosizedaluminosilicates. 4.5Thesignificanceoftheinsitudata

Basedontheconclusionoftheprevioussection,theimportanceofstudyingthesesystemsinsituisapparent.Theadditionoftheinsitusizedataresultedinthemodelsimprovingfromexplaining86and88%ofthevariance,to91and98%,respectively.Thesecomplexsynthesissystemscannotbefullyunderstoodwhenalltheanalysisisperformedaftertheformationoftheparticles.Criticalinformationaboutformationconditionsandparameterscomesfromtheformationprocessitself.TheSAXSresultsalsodemonstratedsomemoregeneralizedprincipalsofparticlegrowthandnucleationthatmaybeapplicabletoothergeochemicalsystems.4.6Multivariateapproachtoexplaininggeochemicalsystems Alongwithutilizinginsitudata,amultivariateregressionapproachhasallowedforthedescriptionforthesystemasawhole.Usingthissystematicmethodologycreatestheopportunitytoexploreunaddressedquestions.Modelingpredictsthatimogoliteisenergeticallyfavoredtoformwhenprecursorparticlesarelargerthan4nm,butithasalsobeenshownexperimentallytobeunabletoformwhenthehydrolysisratioisunder1.5.Usingthismultivariateapproach,assumingcorrectquantificationandexplanationforthefactorsinfluencingthesystem,wecanaskaquestionsuchas,whatwouldbeexpectedtodevelopifprecursorparticleswerelarge,butwereformingatlowhydrolysisratios?Assumingastartingconditionsimilartootherliteraturestudies,andgivenastandardparticlesizeof4nm,whichhasbeenfoundrepeatedlyduringinsituinvestigations,themodelwouldsuggestthatevenwithalargeparticlesize,imogolitewouldbemostlyunabletoformatlowhydrolysisconditions.Conversely,givenasimilarsystembutwithahighhydrolysisratioandsmallparticles,whereproto-imogoliteshavebeenfoundtobestructurallyunstable,butalsoshouldbeenergeticallyfavorable,themodelpredictsthatminimalproto-imogolitewillform.Thisleadstotheconclusionthatthehydrolysisratioisastrongercontrolonthesystemthantheparticlesize,whichmaygiveinsightsintotheformationprocess,andrevealsthestrengthofthissystemsapproach.

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5.ConclusionsWeusedasystematicsynthesisprocedurecombinedwithasuiteofcomplementaryanalyticaltechniques,assessedusinglinearcombinationfittingandanalyzedwithmultivariateregressiontechniquestodeterminetheinfluenceofstartingphysicalandchemicalconditionsonthefinalphasecompositionofnanosizedaluminosilicateproducts.Althoughsynthesismethodspreviouslyreportedtohaveformedallophanewereused,theonlydefiniteproductsproducedwereimogoliteandproto-imogolite.Usingasystemsapproachallowedustoexaminehowconcentrationofstartingreagents,elementalratios,hydrolysisratios,andprecursorparticlesizesallsimultaneouslyandindependentlyinfluencedtheproportionofeachendmember.Weproducedlinearmodelswithstatisticallysignificanthighpredictivepowerthatquantitativelydescribedtheeffecteachfactorhadonthephaseabundanceofeachendmember.Twodifferentdatasetsproducedlinearmodelswithsimilarcoefficientsandidenticalconclusionsaboutthedirectioneachparameterdrovethesystem.Themodelsaccuratelyrepresentedwhatwasfoundexperimentallyinseveralliteraturestudies.

Thesemodelscanhelpanswerquestionsaboutnanoparticleformation,andaddressfactorsinauniquelysystems-basedapproachnotyetemployedinthisfield.Thisapproachcanbeappliedtoamultitudeofgeochemicalsystemstoproducepredictivemodelsaboutcrystallization.Usingthesemodelswecanprobequestionsaboutwhatchemicalandphysicalpropertiesaremostimportantduringmineralorprecursorformation,whichcanleadtoinsightsaboutmechanismsandreactions.

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